B737 flight crew reported engine failure during taxi out. Flight returned to gate.

Narrative

After APU start before pushback; the flight attendants informed me that the back cabin temperature was extremely hot. So we pushed back and started up the number two engine. The heat was still excessive in the back; so we started the number one engine. We had an EDCT (Estimated Departure Clearance Time) time of about 25 minutes after push. So we taxied out to a holding spot and called the flight attendants about the heat in the back. They said it was still extremely excessive. The positions of the air mix valves were both showing full cold. But deduct temperature was well over 50°C and the cabin temperature in the back was over 33°C. I called maintenance to discuss possible options. They suggested resetting a couple of circuit breakers. Before I could reset the circuit breakers the number two engine engine fail" light came on and the engine shut down. We requested a taxi back to the gate which we did after running the engine failure checklist. We deplaned the aircraft. We wrote both problems up in the ELB (Electronic Log Book) report. The flight was subsequently canceled. We did an engine start an idle power run with maintenance on board. After turning the right pack on; the temperature continued to spike in the back and on the duct and cabin indicators in the cockpit. The EGT on the number two engine continued to rise as well above 500°C. Maintenance decided to turn the engine off before it flamed out again."