I can't remember whether it disappears outside of combat, or just after a few seconds at full health. The squad doesn't get full health bars. They get little shapes next to their portrait that aren't as clear as they could be due to the colouring. Squad health starts grey, and becomes red. When they are taking damage, their portrait flashes red.
There is no heat management to worry about, so no bar needed.
Health goes down really fast because none of the enemies ever miss you and they focus fire whenever you are out of cover. The red lines that appear on screen should be more noticeable than the little red bar at the bottom of the screen when you are about to die.