Do I have to give a recorded statement to the insurance company in a New York auto accident case?

That's not required, if the insurance company is your own insurance company ... In New York, we have the, No Fault Insurance, so if the No Faults Adjuster is asking you about the case, you can cooperate in fact, you must cooperate with your own insurance company. It's better not to give a recorded statement, and they usually will ask if it's okay, and you can say, "I prefer not to give a recorded statement," and they will honor that usually. If the insurance company if for the other party involved, you should not give a recorded statement, you should not give a written statement, you shouldn't give any statement, you should politely decline to give a statement and say that you're going to consult with your attorney and let the attorney deal with that.