I was rear ended in a car accident and I don't have insurance but the other driver does, can I file a claim?

Yes, if the other driver was at fault, then his insurance should be covering your damages. The question is what the insurance will contest in your trying to make the claim: fault, extent of damages, prior damages. If there was any injury involved, seek treatment and follow recommendations, gather the records, take photos of the scene and vehicles and set a consultation before giving statements.