I don't think the White House was involved at all. I think his fellow soldiers openly expressed his dislike for him, rolled their eyes when they had to salute him, made fun of him while he was in earshot, and made sure that he knew they were with Trump. In fact, I also think his fat body and pompous attitude would make him disliked apart from anything political. This is just him making an excuse for quitting in the face of unpopularity.