Disclaimer: I have no experience in the hiring process or background checks (apart from being on the receiving end). To answer your questions, you were fired, so say you were fired. If you say you quit, and your previous employer is contacted and says you were fired, that won't go over well. There is a space to explain your answer about why you were fired. It might disqualify you, it might not, but lying will. As TVA said, get over your issues. The person observing you is a professional first. If the behave inappropriately, report them, but refusing to comply in this case is a big deal.