Social Security Numbers and health insurance

[Dr. Evil]

I need to sign up for health insurance tomorrow.  I'm going to buy
individual coverage from Kaiser.  On their form, it asks for an SSN,
of course.

Well, Kaiser is not a government agency, and is certainly not
associated with the Social Security Adminisrtation in any way, so they
have no legitimate need for my SSN, and I don't want to give it to
them.  I'm wondering how best to go about doing this:

1. I could make up an SSN and give it to them.  I basically just have
   to make sure it's one that hasn't been taken, and corresponds to my
   real birth county, right?

2. I could say that I don't have an SSN.  Most people in the world
   don't have SSNs.  Would they then require proof that I'm not a US
   citizen?  Or will they just assign me some kind of policy number?
   That would be ideal.

3. I could refuse to give it to them, but I think they would probably
   then refuse to give me coverage.

Any thoughts?  I have absolutely no health problems that I'm aware of,
and I'm not trying to hide any pre-existing condition, and my only
medical records are things which say, "he's healthy", so I'm not
trying to commit any kind of fraud here, I just want my privacy.

Thanks