If you filed a dispute of accuracy of information in your CR, and provided them enough information, such as your name and the account number that THEY reported to the CRA, that is rock solid ID of the account under which the information is being reported by them, and now disputed by you. They reported it, so they can identify it. They are NOT entitled to require additional personal information in order to consider the dispute. Requst for SSN is inappropriate. It smells of their lack of ability to verify, so they are just trying to make you think that they can shift the burden of dispute resolution back to you. That is BS. That is their problem, not yours. Dont give them more than what is required to match your dispute with their inaccurate reporting to you credit file. Your account number.
Did you dispute through the CRA, under FCRA 611(a), or was it a direct dispute with them under FCRA 623(a)(8)?
If your dispute was filed directly with the OC, I would send them this letter:
"I filed a notice of credit reporting dispute with you on (date). In that dispute, I identified myself, and the information being disputed. I identified the number of my account with you, and thus provided sufficient detail for identification of the information being disputed. FCRA 623(a)(8), and 16 CFR 660.4.
" You have now requested that I provide to you certain items of personal information that are NOT requried under the FCRA dispute process. I respectfully decline your request.
"I have no obligation to provide additional, personal information to you beyond that needed to identify myself, the account that is subject to the dispute, and the specific information under that account that is being disputed.
"The date of my initial notice of dispute to you remains as the 30-day date that sets your requriement to complete your investigation, and then report the results of your investigation back to me within 5 days thereaftr. Your recent request for further disclosure of additional personal information not mandated under the FCRA dispute process simply has no relevancy to your legal invesitigation and reporting requriements.
“Should you be unable to verify the accuracy or completeness of the disputed information wothin your 30-day investigation period,, then you are additionally required, under FCRA 623(b)(1)(E), to promptly notify the credit reporting agencies of deletion of this information from my credit file.”
Some of the guru's can give you more information as to what your next move should be. 
Starting Score: 550
