KBA (knowledge based authentication) is what that question-answer type of system is called. Typically companies don't implement it as the sole form of identification for the exact reasons that you mentioned.

I found it very surprising that Square doesn't provide an alternate method for identification, it's mind boggling even. Something like Instant Bank Verification (similar to what mint.com or venmo does) would be a good alternative and carries no additional overhead for Square.

If no response try the BBB and find one in your area to file a complaint so that their management has to answer you.