(this answer is specific to US laws and regs for domestic flights)
No, there are not. And, in fact, it is perfectly legal to fly with ALL of the toilets broken and unusable. No restrictions on flight time, either. The airline is required to notify all passengers that the flight has no working toilets. (I'm not sure if any compensation is required for passengers electing to not board the flight).
The Department of Transportation does provide the following in their Aviation Consumer Rights rules:
- (in the section discussing tarmac delays): While the aircraft remains on the tarmac lavatories must remain operable and medical attention must be available if needed.
Although it's obviously not possible to make the lavatories available during a tarmac delay if the lavatory is broken, there are no rules prohibiting the flight from departing if the lavatory is broken.