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

It's not required for you to give a recorded statement, and you do have the right to decline a recorded statement. Of course, with your workers' compensation you have to report the accident to your employer, and an accident report should be filed. However, that does not mean that it has to be recorded, and it's not advisable for you to give a recorded statement to anyone, because anything you say can be used, or misconstrued, against you.