If you tell someone that you accept assignment it means that you will accept the insurance company ALLOWED amount as payment in full. So if the insurance company does not reimburse you their entire ALLOWED amount you can bill the patient for the difference between the allowed amount and the payment.
For example: your charge is $300, the insurance company allows $200 and pays you $150. If you accept assignment you can bill the patient the $50 difference between the $200 ALLOWED amount and the $150 payment.
A non-par provider doesn't have a contract so they can bill the patient the difference between the billed amount and the payment received ($300-$200 = $100) since there is no contract stating they will accept a reduced rate.
You do not have a contract with any insurance company so you can technically balance bill for everything not paid by the insurance company. However, if you tell patients that you accept assignment then I think it is bad form to surprise them with a balance bill.
NY enacted a Surprise Bill Law back in 2015 to protect patients from these "surprises" after receiving care from a non-participating provider.