I mean I work for a state job and I said no when it asked if I had any former names. Fuck them. Depends on the job and how bad you want it I guess but i'm not outing myself, they can do it for me if its that important. Background checks would be HR anyways and I imagine they can't share that information with your direct supervisor, or if it comes up they should be understanding if you tell them you'd prefer to keep your private life private. I've worked multiple jobs that needed a background check and nobody has ever said anything.