Can a finance company require an auto extended warranty in order to give you the loan?

You are correct. The dealer and finance companies are not allowed to require the purchase of an extended warranty.

Your credit score might be the concern. Why not try to finance where you bank or maybe a credit union or through work. The dealer makes big money selling these addons so you are right to be leery.

I've heard of all sorts of gimmicks over the years from sales pitches, esp. in the automobile industry. However, no one can compel you to agree to anything you don't want and an extended warranty on virtually all purchases is a questionable add-on for anyone to agree to with virtually any purchase, incl. a car. As suggested, you should consider alternative sources of financing and/or another dealer. Good luck.