why does a claims adjuster need my ssn?

A primary reason is to comply with the Medicare Secondary Payor Act, Section 111. Liability carriers are required to notify the government of personal injury claims in the event that person is receiving medicare and/or medicaid (or may be eligible for Medicare). I cant get into all the issues with MSPRC, they have a right to be reimbursed from settlement/verdict proceeds for payments they make, or will make that are caused by a negligent person. As a secondary reason, they probably want to run an index search on you for prior claims history.