I don't know how much of a requirement it is. I do believe that legally speaking, the only people that can demand one's SS# as proof of identity are banks and the IRS. Everyone else just uses them because they are a convenient way to assign someone a unique "customer ID" (instead of generating their own, which I guess is a hassle).
At any rate, if you really didn't want to, then I believe you have the right to deny giving it to them, as they aren't a government entitity. But they may say it's "for a credit check" or something like that. If you stick to your guns though, you'll probably not have to at all.
It's only a hassle when they first come up with the software algorithm to create a unique ID to their specifications. After that, it's no biggie.
I refuse to give my SSN except to banking firms, IRS and the state BMV (required for Ind. DL.)