If its all on one ticket, then yes, they will rebook you, and unless your inbound flight was delayed due to something in the airline's control, you would be stuck footimg the bill for the overnight stay.
If they were booked as two tickets, I would think there would be a higher chance of being told you are out of luck.
I cannot speak to JFK's customs lines, but at EWR on the high end, we generally see US Citizens through immigration and customs 1 hour after the aircraft opens the door at the gate, and Non-Citizens will get through by the 2 hour mark. And the average is about 30 minutes for US, 45-60 for non.