B737 CREW HAD AN INTERMITTENT OVERHEAT WARNING FROM THE #1 ENG. WARNING REMAINED WHEN ENG THRUST WAS REDUCED TO IDLE; AND THEN SHUTDOWN. ENG WAS SHUT DOWN PER THE ABNORMAL PROCS CHKLIST AND BOTH FIRE BOTTLES DISCHARGED. AN UNEVENTFUL LNDG WAS MADE.

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ENRTE FROM TRENTON; NJ; TO GREENSBORO; NC; THE MASTER CAUTION ACCOMPANIED BY THE OVERHEAT/DET LIGHT CAME ON INTERMITTENTLY. THE ENG #1 OVERHEAT LIGHT FLICKERED ON AND OFF. THE L THROTTLE WAS REDUCED AND THE OVERHEAT LIGHT CONTINUED TO FLICKER. THE CAPT AND I BOTH GUESSED THAT IT WAS LIKELY JUST A PROB WITH THE WIRING BUT WE WERE OBLIGATED TO GO AHEAD AND SHUT THE ENG DOWN USING THE ENG FIRE/SEVERE DAMAGE/SEPARATION PROC. THE LIGHT CONTINUED TO BE INTERMITTENT AS BOTH FIRE BOTTLES WERE 'SHOT.' I THEN ASKED IF WE WERE GOING TO LAND AT THE NEAREST ARPT. HE REPLIED THAT WE WERE CLOSE ENOUGH TO GREENSBORO THAT WE WOULD BE ABLE TO GLIDE IN. I ASKED HIM IF HE WOULD LIKE FOR ME TO REVIEW THE GOM CONCERNING CONDITIONS THAT REQUIRE LNDG AT THE NEAREST ARPT. HE REPLIED THAT IT WAS JUST A WIRE PROB AND THAT WE COULD CONTINUE TO GREENSBORO. APCHING LYNCHBERG (WHILE THE CAPT WAS TALKING WITH OUR DISPATCH); ATC ASKED ME IF WE WERE GOING TO OVERFLY LYNCHBERG. I ASKED THE CAPT IF HE PLANNED TO LAND AT LYNCHBERG. HE SAID NO; THAT WE WOULD CONTINUE TO GREENSBORO. THE CTLR ALSO ASKED ME FOR FUEL AND SOULS ON BOARD. FROM THE TIME THE ENG WAS SHUT DOWN UNTIL WE LANDED THE CAPT CONTINUED TO TELL ATC THAT WE HAD ACCOMPLISHED A 'PRECAUTIONARY SHUTDOWN.' I CONTINUED TO FLY THE ACFT AND MADE AN UNEVENTFUL SINGLE ENG LNDG ON RWY 23 AT GREENSBORO. I FELT THAT CRM WAS GOOD IN ACCOMPLISHING THE SINGLE ENG PROCS BUT THAT I SHOULD HAVE BEEN MORE FORCEFUL IN MAKING KNOWN MY THOUGHTS ABOUT THE NEED TO LAND AT THE NEAREST ARPT. SINCE ATC KNEW THAT WE HAD SHUT AN ENG DOWN; I ASSUMED THAT THEY KNEW WE HAD AN EMER. I SHOULD HAVE ASKED THE CAPT ABOUT DECLARING AN EMER SINCE INDEED WE WERE NOW SINGLE ENG AND HAD NO FIRE BOTTLES REMAINING. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR ADVISED THE L (#1) ENG WAS EFFECTED. THOUGH FAULTY WIRING WAS DETERMINED TO BE THE CULPRIT; HE DID NOT KNOW WHAT MAINT REPAIR/REPLACEMENT WAS COMPLETED. TO THIS KNOWLEDGE; NO OTHER CREW MEMBER WAS CONTACTED BY ANY OTHER FAA OFFICIAL.