What I can't get my head around here is how these employers know someone has a FB (or any other) account, and what Earthly bloody business they imagine that it has to do with them?
Next, are they going to be asking for your bank card PIN numbers?
Tell them to go fuck themselves! And if they fire you, sue them. Better still, don't have an FB account. And if you do it certainly isn't going to be your "real" name.
A few test cases will soon put a stop to this business and take down a peg (or three) these "employers" with an inflated sense of their own importance.
Such impertinence!