The problem with this objection, thus far, is that it has no content!! For any two propositions A and B, using A to show support for B, could be objected to on the basis that someone could support A and not B. But we know that's not right because propositions can (via some explanation) support other ones.
So, to make the "don't use the flag to support the war case," what's needed is to demonstrate that the explanation connecting the two is wrong (not simply to have someone support the flag and not the war, because that person may be wrong/inconsistent).
Anyway, a simple reason that supporting America implies supporting the war, is that the war will make America safer (Iraq funds and supports terrorism). Or in reverse, the anti-war position of wanting Americans to die is not consistent with being pro-America.