With a truly just cause, the soldiers would be more than willing to volunteer and fight. If the war is not just, then the lack of volunteering soldiers would reveal that the war is not one worth fighting. Voluntary Military Service would be a preventative measure from fighting too many wars while at the same time insuring the effectiveness of the military in fighting the wars it does fight.
From a purely military perspective, morale would be much greater if a soldier could make the conscientious decision that he believes the war he is to risk his life in is worth fighting.
From a social perspective, the citizens of the United States would know for a fact that the soldiers were not giving their lives away for something they did not believe in.
From a religious perspective, this would allow people to join the military that believe in the "just war" theory. Right now, proponents of "just war" should not in good conscience enlist in the military. A soldier has to subject their opinion on whether a war is just or not to the heads of the State. When they join the military, in effect, they are making a permanent surrender of their religious conviction to the decision of the leaders of that nation. For an enlisted soldier, it does not matter if a war is just or not; it matters whether they have been given a command or not. Religious beliefs should not be subjected to the state.
The very idea of "just war" means that in every war there is at least one side that is unjust. The logical conclusion is that at least one side should have people conscientiously opposed to that war. It is very convenient, although intellectually dishonest, to say that our side would always be on the just side in a war.
The benefits of allowing soldiers to decide whether to fight in a war outweighs the detriments to such an unorthodox process. Just wars would be decided based upon the conscience of the people rather than the conscience of a select few. The bar for the case for war would be raised. The responsibility for war would be on the shoulders of those actually fighting it. All soldiers would be allowed to participate with a self-perceived clean conscience. There is nothing to fear in allowing soldiers to decide whether they will or will not fight in a war if the war is just. Unless we fear not fighting in wars all of the time.