I don't think it has to do specifically with clicking to shoot --
I had this same issue with the soldier not moving except it was when I tried to have him climb over a fence. He froze at the fence but had plenty of AP to complete the action. I tried to tell him to move back away from the fence and he would not move at all. I couldn't even get him to rotate in place.
But just like you - I could make him kneel/stand and then everything was fine again.