Should I warn my employer about bad mouthing boyfriend?

Yes. You should defiantly warn your new employer about your ex. Try frazing it in a way that he won't get upset, or worry about you. Try saying; "I believe my ex spread bad rumors about me at my old job right after we broke up, and I fear he might do it again." Just let your boss know so if your ex does spread rumors, he'll know they're false.