The word-game proceeds by a set of equivocations. They say:
Today we tend to think of the conscience as an inner voice, or a form of moral intuition. It’s a little cricket whispering in your ear. Your mind talking to itself. The idea that the conscience is an inner source of knowledge has obvious appeal. If your own conscience can tell you what to do in morally significant situations, you don’t need to struggle with others to arrive at justifiable decisions.
They will later have to face the problem that people in these circumstances don't seem to treat conscience as a source of knowledge but as a source of belief; they will get around this by proposing, in a completely ad hoc way, that people are confusing knowledge with firmly held belief. In reality, of course, when we are talking about 'conscientious objection' everyone recognizes that we are talking about belief to begin with -- the whole point of recognizing a category for conscientious objectors is to take into account the fact that people have strongly held religious and moral beliefs. But what they are trying to do with this apparently arbitrary intrusion of 'knowledge' into the mix is found in the last sentence. You see, when you engage in conscientious objection, you aren't asking other people for permission to opt out of them doing things to you.
The particular thing they have in their sights is the so-called 'philosophical exemption'. Now, it doesn't take any elaborate investigation to show that the adjective here is not being used in a technical or formal way but in the colloquial way in which people talk about their philosophy of life and similar such things. The real point of the exemption is to recognize that people could have moral reasons for objecting that are not strictly religious. But Biss and Biss, of course, decide they will take the term in a full formal way:
Philosophy is not a matter of declaring rigidly held beliefs, but of working out what can be held true in conversation with others. In the Western tradition, going all the way back to Plato, philosophy is based on dialogue. But philosophical exemptions to vaccination laws excuse people from explaining themselves.
Even in the sense in which they are using, this is a muddled bit of reasoning. Philosophy may be "working out what can be held true in conversation with others", but it doesn't follow from this that it specifically requires working it out in conversation with this or that group of people (Biss and Biss have certainly done nothing to show that anti-vaxxers aren't conversing about it, for the obvious reason they can't -- the reason it has become a large-scale problem is that anti-vaxxers converse about it at great length with each other, which has led to this being an actual movement and not just a few random people). And unless you hold that philosophy can't actually find the truth it's supposedly working out, nothing prevents your philosophical conversation from leading to what Biss and Biss, at least, would consider "rigidly held beliefs". And even if they did not, it is an entirely different question whether we should have laws requiring people to explain themselves on the matter, one that can only be determined on political and legal grounds, not on the definition of the word 'philosophy'.
They then introduce the Kantian account of conscientiousness, as if anti-vaxxers have any particular reasons to be Kantians. But the real equivocation is here:
Thorough self-examination might not reveal to everyone the true stakes of a decision against vaccination: the risk of exposing infants, cancer patients, and other vulnerable people who cannot be vaccinated to a life-threatening illness. But Kant’s logic still applies: Acting from a belief system that may run contrary “to a human duty which is certain in and of itself” is unconscientious. Conscience demands that the relatively healthy prioritize their duty to protect the vulnerable from disease.
Note what they do not say, even though someone would naturally tend to assume it: they do not say that the last sentence follows from the Kantian account of conscience given. It in fact does not. Any Kantian account of such matters is not going to look primarily at consequences but at maxims -- roughly what in ordinary conversation we call intent. Consequences are largely irrelevant in a Kantian account of anything. And because the Kantian approach is to fit maxims to moral law, to act with intent that is appropriate to being a rational being as such, there is no sense in which we 'prioritize' duties in a Kantian account -- either we have it as a duty or we don't, either this is one of the circumstances in which we must do it as our duty or it isn't. What they are doing is using a Kantian account to argue that we should examine our motivations, and then splicing a different account of conscience onto it to get their preferred conclusion.
This becomes much more obvious when one recognizes that their repeated insistence throughout on conversation is not particularly consistent with Kantianism. There is indeed a very fundamental sense in which conscience can be considered social in Kant's account, but it is not in the sense of "working out what can be held true in conversation with others" or, as they later put it "work out what counts as fulfilling our duties to others with everyone in our community". (In moral matters that would be what Kant calls heteronomy, and is very much not consistent with what he would regard as conscientiousness.) It is in something like the sense that when you act in moral matters you are effectively doing so as a rational being, and therefore are making moral decisions for everyone, not just yourself, so your moral decisions have to be suitable for everyone. But a Kantian account would also very much have to say that you should sometimes stand your ground on a moral matter, no matter what anyone else might say.
How would anti-vaxxers be assessed in Kantian terms? I don't know. Since it matters a great deal what they are actually intending, and anti-vaxxers probably intend quite a wide range of things, it would depend on the case. Protecting the vulnerable from disease is universalizable, so it would certainly be a duty. But it would also certainly be an imperfect (i.e., incomplete) duty, not on its own telling you exactly what you have to do in order to do it. Vaccinations are not universal rational options that all rational beings have access to; it's in principle possible that there could be medical methods massively more effective than vaccines, so it's an empirical matter whether they are among the most effective ways to protect people from disease; from both of which it will follow that there is no specific duty to vaccinate. It still may be that fulfilling your duty to protect the vulnerable requires vaccination; but this, of course, is precisely what is usually at issue in this case. (Mavis Biss is a quite competent Kant scholar; nothing directly attributed to Kant in the article is wrong. Perhaps Mavis Biss has a much stronger view than usual about what is involved in imperfect duties, or of the way in which community functions in a Kantian context, or something else. That is possible. But the problem is that we seem to move in and out of a Kantian context without warning or indication that it is even being done, which is, again, equivocation.)
Thus Biss and Biss are pretty clearly not appealing to Kant because they are proposing (here, at least) a Kantian view of the anti-vaccination movement, but to show -- well, I don't know what; perhaps that they are more thoughtful than the rubes. It's unclear to me whether they think that Kant is the source of the individualistic view of conscience that they are opposing -- they say a few things that possibly could be interpreted that way, but really Kant is just thrown into the argument, and that whole part of the discussion doesn't seem to contribute anything to the actual argument beyond the self-examination point, which didn't need his authority to be made.
When we step back and look at the argument overall, it becomes clear that they take exemptions for conscientious objectors to be exemptions "from our obligations to others"; this is a very serious misunderstanding of conscientious objection exemptions, which are given so that we can fulfill our obligations to others, and which do not preclude conscientious objectors from fulfilling their obligations to others. This is obvious in every other case of conscientious objection; it's not suddenly untrue here. And all of the word-games in the article mean that Biss and Biss never really argue that the anti-vaxxers are doing anything wrong. They don't argue that our obligations actually must be fulfilled by vaccinations in particular; they don't argue that anyone actually has the account of conscience they are opposing; they don't in fact argue that their account of conscience as having the particular social component that their argument requires is a correct one. Now, I'd have no problem with this, except that they made such an extraordinary fuss about the claim that philosophy was "not a matter of declaring rigidly held beliefs, but of working out what can be held true in conversation with others", and yet here we find them very much not engaging in any kind of dialogue or conversation with anyone, doing nothing but forcefully declaring their beliefs -- whether rigidly held, I cannot say, but there's nothing in the article to make it possible to say that they are not.
Vaccination is one of the greatest medical discoveries of all time, and the value of what it has contributed to the health of the human race is perhaps rivaled only by improved sanitary conditions for the sick and antibiotics. But vaccination in general is also quite intrusive and, what is worse, somewhat indefinite in the extent of its intrusiveness. Talking about conscientious objection to vaccination is often made to sound like it's protesting over getting a small handful of doses, but the CDC's recommended vaccination schedule involves various vaccines adding up to over a hundred doses over a lifetime. (Most people who would characterize themselves as having had all their vaccinations really mean that their parents made sure that they had all the vaccines legally required at the time for schoolchildren, and perhaps a few others afterward. More zealous people do regular flu shots, and probably a few others as they happen to come up for traveling or in medical consultations or what have you. No doubt a few people are as entirely thorough as they can honestly be. But we could add any number of other vaccinations to the list, depending on any number of medical discoveries or newly recognized threats. There is no particular number that is the number of vaccines we should have.)
Vaccination also requires a fairly significant amount of trust in doctors, and in vaccine supply lines, and in medical researchers. Critics of the anti-vaccination movement tend to be good at arguing for the value of vaccines in the abstract; but the movement has largely built on a loss of trust by certain populations in significant portions of the actual medical establishment.
And even if none of this were true, large-scale medical programs by their nature generally need voluntary participation anyway; depending on the kind of program, it's sometimes just voluntary participation of the community generally, but vaccination doesn't work that way. It's the sort of thing you might want to be required; but actively imposing it on people against their will is not actually going to help things in the long run. The standard way of getting around this problem is exactly the one we use: make it legally required, but allow fairly generous exemptions. That way most people will do it both freely and as a duty. One can improve it even more by making it even easier to keep up with vaccinations; that's basically what we do with flu shots. Misinformation, of which there is certainly a lot, can be fought by better information campaigns. It's entirely possible to be insistent on the importance of vaccinations while also recognizing that honest and decent people can have worries that need to be addressed; and respect for conscientious objection is a recognition of the importance of consent and patient autonomy, which are pillars of modern medical ethics. Conscientious objection is entirely the wrong point about which to worry.
(I should perhaps note that Biss and Biss never actually give any legal recommendation. There's a sort of implication, at least an apparent one, that they think philosophical exemptions, or most philosophical exemptions, are illegitimate, and one could read the article as suggesting that they should be eliminated. But the authors never actually say this, and an alternative reading would be just that they are arguing for a regime of persuasion rather than legal coercion. If that's the case, it's not obvious what most of their argument does toward that end, and the argument they give seems actually more subversive of the notion of medical conscientious objection than that would suggest. But the argument given is at its strongest in the insistence that conscience is not purely individualistic but communal, and if you emphasized that aspect of it, you could very well argue that the argument's practical import is really that these things need to be 'discussed' (in some way).)
ADDED LATER: Related to this is this recent article on how much easier it is to get parents to agree to vaccinations if you just bother to answer their questions patiently and start giving them information early and well before the vaccination times come up.