I don't know you or your mother but from what you've posted I think you're misinterpreting what your mother said.
It's not unheard of, and can be common, for employers to check the facebook account for people they are thinking of hiring. They do it with the rational that it gives them a better idea of what kind of person the candidate is. Typically they are looking for someone who is posting a lot of unsavory things (such as lots of racism) or posting pictures of them drinking/using drugs.
What I think your mom is worried about is that a bigoted employer might see that you are gay and not hire you because of that.
And I know right now you're probably thinking "I wouldn't want to work in a place like that." But the truth is that it might not be your boss or even a coworker making that decision. It could be some HR drone that's told "check out this profile and let me know if you think we should hire them."
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