First of all, it is always wrong to formally cooperate with an evildoer, where to "formally cooperate" is to share an intention in virtue of which the evildoer counts as an evildoer. But in an unjust war, the leaders of one's country are evildoers (perhaps non-culpable ones, but that doesn't matter), and they are evildoers in part because they intend the deaths of enemy soldiers. Thus the ordinary soldier in intentionally killing enemy soldiers is formally cooperating with evil. Moreover, even if the cooperation were not formal but material (i.e., the cooperator did not share any of the evil intentions of the evildoers, but nonetheless materially contributed to the evil), given the fact that few evils on the face of the earth are as bad as an unjust war, the presumption against such cooperation would still be almost indefeasibly strong.
This seems to me to underestimate just how easy it is for leaders to violate the conditions of just war. If you take Aquinas's position on the question, for instance, a leader can fail to war justly even if the cause is just, the means are just, and the authority with which he does so is just, if he fails to be virtuously disposed in his activities on the question. It is possible to do the right thing for the wrong reasons, and this violates justice just as much as doing the wrong thing. If we assumed that obeying orders automatically involves formal cooperation, it would be generally immoral to be a soldier, because one would constantly be obeying superiors with defective intentions. Genuine justice is difficult, particularly in matters as serious as those that usually lead up to war; and its difficulty makes it rare.
Further, soldiers needn't, in obeying orders, be formally cooperating with their superiors, in the sense Pruss notes. They are cooperating in some sense, of course; but even obedience does not imply that they "share an intention [with the evildoer] in virtue of which the evildoer counts as an evildoer". I'm inclined to think, actually, that this fact is one of the things that makes the view Pruss is arguing against a common view. We know the sort of people many soldiers are; they are often friends and family, who have often signed up for the sort of obedience involved in military matters not because of any taste for war but because they want to protect their people.
In a later argument, Pruss says:
For in a just war, the state permissibly authorizes the killing of enemy soldiers. But it is wrong for the state to authorize the killing of people who are doing nothing wrong. Hence if the soldiers on the unjust side are doing nothing wrong, it is wrong for a state with justice on its side to authorize killing them. And that is absurd.
It never enters into most just war discussion at all whether the soldiers on the other side are doing anything wrong or not, and for good reason. You can in principle have a just war in which all the soldiers on the other side are of sterling character, just and good, but their leaders are not; for instance, if the soldiers are being effectively deceived by those leaders (as Pruss notes later). In such a case, war may be a just option, but the fact that the soldiers on the other side are good, decent people is a tragedy, not an impediment. More importantly, in just war, the state does not permissibly authorize the killing of enemy soldiers; it authorizes the use of force to protect the common good, even to the point of being lethal. That's a very different thing. Authorizing the killing of enemy soldiers is unjust massacre, not just war; leaders instead authorize soldiers to protect the good of the people with what force may be required. And it is important to note that even if the leaders unjustly go farther than this the authorization to use what force required to protect the common good is still legitimate. Even granted that disobedience is required (and I would agree that it sometimes and perhaps even often is), soldiers still must obey those orders that are reasonably required to keep their nation from being invaded or harmed, because authorizing that does fall within the realm of legitimate (and just) authorization.
My point is not that soldiers are justified in obeying just any decent orders (in the sense Pruss notes above), since I don't think they are, but that neither cooperation nor the innocence of the enemy provides a strong enough argument to reject the common view against which Pruss is arguing. Obviously a lot will depend on particular circumstances -- but that is precisely why the common view is not easily refuted.