The patient can submit a claim to his insurance company if he paid cash. He will need a statement from the doctor's office with all the correct coding to submit the claim.
The insurance is still only obligated to pay the allowed amount. Some policies have no out-of-network benefit so they would pay nothing. If there is out-of-network benefit the patient's deductibles and co-pays still apply
So if the service was $100, he went to an out-of-network provider and paid $100, the insurance would pay the out-of-network amount less any deductibles and co-pays. That could be calculated on U&C amounts, or allowed amounts. And this will vary from insurance to insurance.