I decided this summer to put a lesson on conscientious objection in my Ethics courses this term, and, having refreshed my acquaintance with a lot of the literature, I am increasingly wishing I hadn't. The whole field is just a completely embarrassing mess, and despite lots of jabbering, very little serious work has been done that's not patchwork and question-begging. Fortunately, it's not like discussion of conflict of interest (which I've noted before is a relatively recent term that, despite its importance, has no adequate philosophical account)-- it's been around long enough that it has accumulated bits and pieces of serious thought, and advocates of this or that form of conscientious objection have often had to be fairly precise and reasoned about their approach -- but looking at the areas in which conscientious objection has come up as a particularly significant issue (military service, vaccination, taxation, abortion, euthanasia, gay marriage, gender transition, emergency contraception, etc.), it's quite clear that a lot of the work being done by philosophers consists in making things up to get the answers they want, because the work certainly doesn't suffice for a general account.
Nonetheless, we can say a few general things. Conscientious objection, generally considered, is when a person operating in a matter for which they are responsible refuses to cooperate or comply with an action that they judge to be inconsistent with the ends of that responsibility; in addition, we are not usually considering just any kind of action but one that in some sense is normative (usually due to some law, institutional policy, or proposed professional standard) which gives us the 'objection' part. It is not a necessarily a unique kind of action; an action that is conscientious objection in one situation might not be in another, because in fact it is ordinary conscientious action in a situation that has created a moral dilemma. (This is, unfortunately, often overlooked in discussions of conscientious objection in the literature.) There are several questions this raises -- for instance, how this works out given different kinds of responsibility, or different kinds of normativity (professional standards, for instance, are abstractions from conscientious action of professionals that are then articulated and endorsed by specific professional organizations for a wide range of reasons, and do not at all operate in the way that law does), or interaction with other rights of various kinds. Likewise, the standard ways of accommodating conscientious objectors -- special exemptions or alternative services -- can come in a wide variety of forms.
To object conscientiously doesn't need justification. (This is also often forgotten in discussions.) If you are engaging in conscientious objection, you are refusing to comply or cooperate with something on the grounds that your best judgment indicates that it is a morally bad thing to do. There is no need for any further justification for resisting compliance beyond this. Likewise, you don't need to have reasons to 'permit' or 'allow' somebody to avoid doing something bad, so when people talk about conscientious objection being 'permitted' or 'allowed', they are tendentiously talking about something else entirely, namely, what should be done in the way of coercion and sanction by people who think this best judgment is false in matters that are deemed for some reason to be of major importance to the broader population. (It's not particularly controversial that, at least as a moral matter, people should usually not be forced to do things they consider inconsistent with their moral responsibilities in matters of little or no importance to the broader population.) People in general agree that there have to be some limits, but disagree about exactly what they are, because it matters a great deal how one understands conscientous objection itself, and, as I said, there is no generally accepted account. There are four kinds of arguments that typically come up in the context of trying to pin down limits to what can count as conscientious objection; as far as I can tell, none of them have really been examined and discussed as they should be.
(1) Primacy of Conscience. Usually when you get people who are trying to explain conscientious objection, appeal to conscience comes up. On certain views, conscience is either the primary moral faculty or intimately connected to it; because of it there are things you cannot "in good conscience" do, so there is at least some presumptive moral obligation to act according to your conscience. It's disturbingly common for people, especially bioethicists, to note that there are different accounts of conscience and then move on, as if you could then just ignore the subject of conscience itself. But of course, the point of talking about the primacy or authority of conscience is to talk in a more specific way about the primacy or importance of people. When you force people to do things that they are claiming are against their conscience, this is a paradigmatic instance of attacking a person. It is the sort of thing that in other contexts is associated with police states, oppressive regimes, and certain forms of psychological torture. So if conscience is the foundation for thinking about conscientious objection, it is not a minor issue what conscience is, because it will be intimately connected with what it means to be unjust to, and oppressive toward, a person. Sometimes people will also try to use a particular account of conscience to try to gerrymander a way in which one kind of conscientious objection (say, to participating in military operations in a war) will count but another kind (say, to vaccinating one's children according to a vaccination schedule that is not trusted) won't; but this generally seems not to yield anything useful.
(2) Necessity of Conscientiousness. In actual fact, however, I suspect the more common practical foundation for conscientious objection is not the primary concept of conscience but the secondary concept of conscientiousness, which is surprisingly little discussed. This is not a minor failure, either, because if we look at the kinds of situation in which conscientious objection can become important -- citizenship, civil service, professions, and the like -- they are all cases in which it is generally important to be conscientious. Opponents of a particular conscientious objection in a professional field, for instance, often draw on the primacy-of-conscience approach in a way that lets them treat the matter as an opposition between 'personal' standards and 'professional' standards. But this is quite clearly illegitimate if we focus on conscientiousness; having professional standards (for instance) presupposes conscientiousness, so the possibility that a proposed professional standard is creating a common occasion for conscientious objection is in itself a reason for thinking it is a badly formulated professional standard. Of course, someone could have an idiosyncratic and dubious sense of what conscientious practice is, but the divide between personal standards and professional standards is an artificial one when we are talking about conscientiousness.
(3) Presumption of Liberty. Because we can all recognize that we are all better off not being constantly forced to do things we think are bad, it seems reasonable to say that we shouldn't introduce coercion and sanction into a matter unless we have to do so; but, of course, the question of note is what makes for a situation in which we 'have to do so'.
(4) Additional Value of Accommodation. Accommodation of conscientious objection can have benefits in and of itself. For instance, it can serve as a wall against any tendency toward totalitarianism. These kinds of benefits often get mentioned, but are often not developed at great length.
A great many questions are just never answered properly. In teaching, there's not generally a problem with cleaning up a field that's become a bit messy, but when it's in such a complete disarray as philosophical discussion of conscientious objection, it's hard to know where to begin.
