sorry to hear. But isn't Ohio a right to work state? Secondly say you fight is and are found in the right and they offer you the job again do you think they will really want you there and would you really want to work there after having legal issues with them?
They would watch you closely wait for the first mistake and fire you.
I say chalk it up as a life lesson. Keep the paperwork and incase you find another job that ask for the same info and that way you will have your bases covered.