Meh. Protecting your name is only for scaredy-pants people for whom it actually has any power - like in the days of yore where you wouldn't tell it because you were afraid of magic...
Then again, I guess with no proper social security number, and the irrational fear of having one you seem to suffer from over there, you have to fall back on your name (and "proof of address", lol, hellooo stone-age) so I guess it actually has some power.
Disclaimer: the above paragraphs were just taking the piss out of you. Well, okay, maybe not entirely, but imagine the cheeky things being said in a diplomatic way. In the end, I guess it depends on whether you think someone is "after" you. If you think someone is - don't join the facebook group. And don't tell your bank account number to the nice man from Nigeria, either. Oh, and if you're going to let people post pictures of you on fb, which you maybe wouldn't want future employers to see, for the love of god use the privacy options, and hope they don't snoop around too much.