If you play the mission and 4 of the 5 players die, the last player can't control any troops or the hostage. That sucks because you can't complete the mission. We tried to push the hostage to the exit point, but after 30 meters, the lobby crashed.
It would indeed appear that the player taking the hostage has to stay alive.
Depending on the mission, it's recommended to keep the hostage and the player at a safe distance.