It makes perfect sense: if a soldier agrees with the war, and I disagree with the war, why should I "support" him? If Country A invades Country B for no reason, and the world (and myself personally is outraged at that), why would I cheer for Country A's troops?
Okay; there is a difficulty in this, because saying you don't support the troops is like a guaranteed way to have your patriotism questioned.
I understand the point you're making. But the thing that you have to keep in mind is that the troops themselves have no say in whether or not they go to fight, or who they go to fight. They follow orders. While I understand what you're saying philosophically, about "sit in a jail cell" if you disagree with the premise of what you're fighting for, this is really kind of ridiculous if you think about it. These aren't drones that simply exist on principle - they're people who have families and lives to support. Principle is a great thing, but if it means costing you your job, welfare, and future (a dishonorable discharge from the military is just about a guarantee that you're going to have a very hard time getting a job in the future), it becomes a little more difficult.
You can support the troops without supporting the war because the troops didn't decide to be there. They're there, and they're following orders, and they don't really have a choice in the matter. If they support the war as individuals, yes, feel free to disagree with them. But ultimately, the people to hold accountable for these opinions are not the troops, but the leadership that instills these beliefs, and orders troops to follow through on them.