Your instincts are good in trying to preserve the privacy of your social security number. Lawyers will ask for all kinds of stuff as part of lawsuits or collection activity; the burden falls on you to object if an item seems to be irrelevant or overly intrusive. You can answer the demand by providing the last four digits of your SSN; that would enable the other party to prove or disprove your connection with a bank account, for example, without allowing them to go on a fishing expedition. If there is a judgment you don't expect to be able to pay, you may want to talk to a local lawyer about bankruptcy, which may make it possible to remove the lien from your house.
Disclaimer: This site exists to provide information only. It is not legal advice. Answering your question does not create an attorney-client relationship. I am a Massachusetts lawyer. Any information provided on this site does not, except as explicitly stated, imply familiarity with laws or procedures peculiar to your state which may differ from those where I practice.