No, there's no rule in FAA land. The FAA AIM recommends having them on below 10,000 but no rule.
4.3.23.c.
The FAA has a voluntary pilot safety program, Operation Lights On, to enhance the see−and−avoid concept. Pilots are encouraged to turn on their landing
lights during takeoff; i.e., either after takeoff clearance has been received or when beginning takeoff roll. Pilots are further encouraged to turn on their landing lights when operating below 10,000 feet, day or night, especially when operating within 10 miles of any airport, or in conditions of reduced visibility and in areas where flocks of birds may be expected, i.e., coastal areas, lake areas, around refuse dumps, etc. Although turning on aircraft lights does enhance the see−and−avoid concept, pilots should not become complacent about keeping a sharp lookout for other aircraft. Not all aircraft are equipped with lights and some pilots may not have their lights turned on. Aircraft manufacturer’s recommendations for operation of landing lights and electrical systems should be observed.