To "eat s--t" is literally a reference to the behavior of dogs, some of whom do this from time to time. To say that someone "eats s--t" is just one of a number of ways of vaguely insulting someone by associating them with things, animals, body parts, etc. that are considered shameful or disgusting. "Eat no s--t" means "don't behave in a way that would make other people say 'he eats s--t,'" "don't behave in a way that makes people describe you in vulgar or obscene says." The motto could just as well be "don't be an a--hole." It might carry the idea of "I don't take s--t from anybody" meaning "I don't let other people boss me around if they don't actually have authority."

that means this is not your job.

Hi Cyanstar, "Eat no shit" is a very informal slang term for "don't eat junk." Its usage is not too common, since the word "shit" can be considered a curse word (and "eat shit" is an insult).