When an insurance company goes out of business, what happens to the medical records?

I don't know, but interesting question.

Check through your policy/claims items. They may have sent you something with the notice you received from the state when the company closed. The insurance commissioner from your state would be able to give you a better answer. Most likely, the records have been placed in storage somewhere and will be kept in compliance with state laws (7-15 years from case settlement). Do an internet search with "commission of insurance state" with "state" equaling your state. Once there, do another search for "insolvent company". If that doesn't work, find the contact information page and send an email with some specifics. Good luck!

Depends - if the company is acquired by another, the successor company would get the records. But with insurance, you are talking about claim records, not clinical "medical" records. The providers retain those.