. . . . When I push in the cc button, the CRUISE light comes on, then goes off. It will not come back on until I turn off the car then turn it back on, with the same result. . . .
Do you have other diagnostic codes such as P0581, P0582, P0586, or P0594? These are codes associated with abnormalities in the speed control switch circuits and vacuum, vent and solenoid control of the control servo. If any circuit problems are detected the PCM will NOT allow the speed control mechanism to be activated. This is a manufacturers safety design to preclude any possibility of run-a-way throttle on the vehicle.
If you have no other specific speed control codes then the system is sensing excessive resistance in a speed control circuit and disabling the speed control operation as noted. Disconnect the battery cable. Remove connector C3 at the PCM. Carefully clean the pins and mating receptacles at pin locations #5, #7, #12. These pins and receptacles will look satisfactory but clean accordingly. See attached images.
I had the exact problem you describe on a 2003 Neon speed control system. Start the engine and start forward motion. Push cruise control button and momentary engagement and then system shut down. I removed connector C3 and cleaned the pins. Problem resolved.