If you live in Florida and you were rear-ended by a car that was hit by another car and you have no insurance will the car that caused the accident have to pay for your car?

In theory, yes. I don't know about in FL, but in NC, it's my understanding that if you don't have insurance, it doesn't matter who's fault it was, you're liable. In some cases, you may even be liable for the other car's damage...