That is incorrect. The President is hired by the Board of DIrectors. If the shareholders vote against the President remaining a member of the Board, that doesn’t void his employment contract as President. It just would mean that he’s President but not a member of the Board.

Unless the By-Laws are written such that the President is a member of the Board ex officio — by virtue of the office — which means that shareholders don’t vote on it at all because it’s part of the institutional structure. Lots of non-profit organizations have that in their by-laws.