Let's say that Satan is, by definition, the worst possible thing. If something is the worst possible thing, then it not only must have lots of bad properties, but it must not have any perfections; it must be the kind of thing that could not be made any worse than it already is. If it had a perfection, it would be better, not worse, than a thing that lacked that perfection, and thus would not be the worst possible thing. Next, we adopt the Ontological Argument's premise that existence is a perfection. And the conclusion swiftly follows: Satan must lack existence.
The problem with the argument is that if existence is a perfection the worst possible thing cannot be such that it must lack it. Or, in other words, what we usually mean by 'worst possible thing' is 'worst thing out of all possible things'. And since possible things are things that can exist, the worst possible thing is simply going to be the worst thing consistent with the perfection of existence. There are other things we could mean by 'worst possible thing'; we could be using the term 'possible' in some nonstandard way to include things that cannot exist. But in either case, there is no particularly significant link between this argument and the ontological argument: the argument isn't parallel and the proponent of the one isn't automatically committed to accepting the other.
Aikin and Talisse go on to commit a logical mistake here:
If one accepts the Ontological Argument for the Impossibility of Satan, one must hold that there are some evils that, in virtue of not existing, are worse than evils that do exist. Consider a case of evil – say, the kidnappings in Cleveland, Ohio. An implication of our Ontological Argument for the Impossibility of Satan is that the morally identical copy of those kidnappings that might have happened in Pittsburgh but did not, are worse than the ones that occurred in Cleveland.
I'm not sure what 'morally identical copy' is supposed to mean here, since actual circumstances are morally relevant, but it doesn't follow from "there are some evils that, in virtue of not existing, are worse than evils that do exist" that "all evils that do not exist are worse in virtue of not existing than evils that do exist"; for instance, there could be evils that presuppose something existing for their evil (and, indeed, the kidnapping example trades precisely on that fact) without the existence itself being evil (because, again, actual circumstances are morally relevant -- for instance, whether something is really kidnapping depends on what is really true about its context). This latter point is not inconsistent with claiming that existence is a perfection; put them together and you have a version of the privation theory of evil, which is, in fact, the most plausible view of evil if you accept an ontological argument. (It's the most plausible view of evil on most reasonable presuppositions, which is why it has traditionally been the most common view of evil; but it is especially plausible if you accept the ontological argument.)
They go on to say, about the premise 'existence is a perfection':
And that's the premise driving our Ontological Argument for the Impossibility of Satan, and some version of this premise features in all versions of the Ontological Argument for God's Existence that we know of.
I suppose it depends on how strictly or loosely one interprets the premise, but it's worth pointing out that Anselm's argument doesn't rely on any such premise, if we are interpreting strictly. The closest is the claim that what exists in reality and the mind is greater than what exists merely in the mind. This does not strictly require that existence be a perfection; it is a comparative, not a categorical, claim, and I see no reason to think that there aren't other suppositions on which it would also be true. If, for instance, Gassendi's claim against Descartes, that existence is not a perfection but that without which there is no perfection, is true, Anselm's argument still proceeds as it ever has. (It is true, though, that there is some interpretation of the 'existence is a perfection' premise that Anselm would accept; this is true of virtually anyone of even broadly Platonistic tendencies, whether they accept an ontological argument or not.) It is also worth noting that the ontological argument at the beginning of Spinoza's Ethics does not depend at all on anything that looks like such a premise. And, indeed, all you need to run an ontological argument is a possibility that implies existence, or an idea that implies its corresponding reality; even in Cartesian contexts, 'existence is a perfection' is merely put forward as the reason for being sure that you have it.
But in any case, it is clear enough that we have to be careful about two things here. (1) The notion of a given premise "driving" an argument. It is quite true that there are ways in which a premise might do more work for a conclusion than its co-premises, but conclusions derive from their whole set of premises. And in Aikin and Talisse's Satan argument, it's not merely the 'existence is a perfection' argument that is doing real argumentative work but also assumptions about possibility. (2) Even if we hold that the 'existence is a perfection' premise does the lion's share of the work, and the Satan argument has force, one can't get immediately from the Satan argument to the problematic implications that they think make the premise problematic.