Yes, they have the right to sue you. Normally they will just turn over an amount like that to a collection agency. They cannot place a lien on your car, but they could obtain a judgment against you and then have the sheriff execute on the judgment, which could include seizing your property for sale to satisfy the judgment and costs. This is not likely. It would be better for you to start making small, good faith payments until your situation improves.
Of course you can be sued for an unpaid bill.
The issue is why the insurance company did not pay, as you were still employed and, thus, still insured at the time of the surgery.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
Sign up to receive a 3-part series of useful information and advice about personal injury law.