The evil boss: Capt. Zapp Brannigan (Billy West)
Miserable employee: Lt. Kif Kroker, above (Maurice LaMarche)
The vile task: Being subservient to an arrogant, incompetent human, who also happens to later sleep with the girl he, in essence, marries. (It was technically after his death and resurrection, but that's not the point.)
Why he doesn't quit: Kif believes bigger things are waiting for him, and he doesn't like conflict. In the end he does defy Zapp and escape with his girl through a wormhole.