If it was a bill collector or something similar they would have no interest in keeping their identity secret. You should hung up the moment you found out he/she said 'a company'.
@episcopo: never give out personal information to anyone/anything you don't know. Especially if they're asking about your SSN. It's likely just a phishing scam. Hopefully they didn't get enough information to do any damage.
Yeah agree with Conmused and Vetterli, this is definitely fishy. Hopefully they don't have enough info to do harm.
A simple solution don't talk to strangers, that is what caller ID and voice mail is for and if they are legit they will leave contact info. But I wouldn't worry it sounds like you didn't bite and feed them any info they needed to continue.
Did you call the number back? Probably worth doing just to see who answers.
Even better, block your phone number before calling.
Well, I can tell you that no legitimate company would call you out of the blue and ask to "verify" or otherwise you have release your SSN. Now, if you called /them/ and it was a legitimate service like your utility company, they might have a reason for confirming your SSN (that's how mine does it though, a PIN would be a more secure option than the last four of SSN but oh well), but obviously in the legitimate situations where you are asked for, it, you know they are okay because you called them and they already have your SSN :). I would never give out any information, much less SSN, to anyone calling me – especially if I don't know who they are!
Yeah that does sound a little weird.