Can you, love a sinner but hate the sin?
"Love the murderer, but hate the murder?"
"Love the rapist, but hate the rape?"
Mr.Pink ... please explain how you can support troops that are carrying out a mission that you believe to be immoral, illegal, or for whatever reason you oppose the mission? It might .. just might, perhaps .. be logical if these troops were drafted into the military and forced to perform the mission upon threat of death or grievous bodily harm. These are volunteers; most, if not all, are performing the mission because they believe in it. The remainder are performing the mission because they took an oath to obey their superiors. In either event, the soldiers ARE the mission.
You can no more logically state that you back the troops and oppose the war than you can say that you back the murderer but oppose the murder.