When is the tailgate light supposed to come one? When the brakes are hit? When the lights are on? Both?

 

Your tail lights illuminate when your lights are turned on. Both brake lights and the center high mounted brake light should illuminate when the brakes are activated. Your turn signals should also flash when activated. What's wrong with your Custom Cruiser?

 

The light on the tailgate doesn't come on - either with the lights on or when pushing the brakes. We aren't sure if it is a "running" light or a brake light. From our research, we think this is the third brake light. Are we correct?

 

Yes. That is the Center High Mounted Stop Lamp. These were required on all cars by the govenrment starting in 1986. It should illuminate with your brake lights.

Here's a pic:

 

If you have already tried changing the light bulb, then you may have a bad light socket or a wire issue... Something as simple as a ground wire to the tailgate or light itself could be the problem...