I'm a filipina nurse working here in the UAE, No... the California BRN doesn't need a SS # when you register for their NCLEX exam. you can take the Nclex without it, But.... when you pass the Nclex you can't get your license without the SS #, then your papers will not be processed by the INS, so you can't file for the so called I-140 for green cards, you'll get stucked, but this will not affect those with CGFNS certificates, they can still have their INS I-140.
I'm also bound for LA, my agency in LA, advised all their future nurses to get and pass the CGFNS, so they will be able to get their visas soon, luckily I have my CGFNS, so now i'm waiting for my packet 3 which will be out soon this month ( after 4 months of filing). These changes form the Ca BRN came out last July.
I hope I have enlightened you a bit with regards to Ca BRN, so if you still don't have your CGFNS, start applying now, if you have it, then well and good, keep rollin'.....