The Shulchan Aruch (CM 70, 1) says that a loan with a shatar is משובח יותר do the poskim say the same about an employment agreement?
It is definitely the correct and smart thing to do. The more that is written, the more it will minimize misunderstandings, aggravation and fights. The Chofetz Chaim (Ahavas Chesed 10-13) writes this regarding hiring workers and the obligation to give them their full wage, that if things aren’t spoken and worked out beforehand, it can cause fights and other problems. He doesn’t say that it has to be written, however the idea is the same.