Is there a law that will force them to pay it? No, there isn't.

But if you still have the letter stating that your coverage will continue till the 31st, you have a pretty good chance of prevailing in small claims.

Also, if your last day of work was 10-13 you are still well within the legal limit to elect COBRA. You could elect COBRA for the first month which would at least get the bill paid, and then use your letter to get reimbursed for the COBRA premiums in small claims.