Required? No. Helpful? Quite possibly.
Veterans, like everyone else, need an advocate that is aware of possible needs during both war and peace. There have been a few political figures and President's who have gathered insight from their military experience.
On the flip side to this, a career military person has been engrossed in a culture that is different from the civilian culture. If someone becomes a great General, then they are exactly that; a great General. They understand the things that helped them reach that position. It might be more difficult for the same person to make a great President. I think that Norman Schwarzkopf knew this.
I think that the President should definitely listen to their Joint Chiefs of Staff though.