I don't see how they ought to have rights to such a thing. Having a screen name ensures your anomynity and privacy on the net. Your usage and antics while on the net are your own - it's part of your personal life.
I don't see what grounds employers have to pull that one.
I Googled my screen name, and it came up with 432 results, most of which are from this forum. In fact, I would say that over 96% of them are from this site. The others are from Deviant ART, Myspace, and from the obscure compliment given by a friend who is a member of several other forums.
Hell, it's to the point where I can simply type in my screen name and be taken immediately to my Member Profile here.