part one.
Can blue cross blue shield of illinois deny one health insurance if
one lacks a social security number?
part two.
Can a company like blue cross blue shield of illinois retrieve all of
your health history, if scattered around different doctors?

Blue Cross can deny anyone coverage for any reason. If you read the
fine print in their disclaimer they are very strict. If you are
without an SSN they can deny you as well however it is unlikely. You
may not have an SSN for many reasons and generally that is not held
against you.

As for the health history, they can retrieve information from many
sources. It would be wise to disclose as much information you have on
your health history as possible to ensure coverage is not removed
months down the road due to misinformation.

Withholding information about your health history can not only result
in cancellation of your policy, but in legal action to recover funds
that were paid in the past, if you obtained coverage based upon false
information.