Interesting that when I had to buy medical coverage today when my COBRA lapsed I was at first denied because I declined to provide my national identification number (aka SSN). The moment I added it, hey presto, they accepted my application and gave me a policy.
A not-so-subtle way of forcing you to provide SSN. As far as I know, that's quite illegal if you're not a government agency. I must look up the law on the matter.
Unfortunately, the government doesn't seem to enforce this, hence the huge rise in identity theft. As a victim of this myself, I'm pretty ticked off about it - and the impossibility of getting a new SSN issued.