B767 CREW HAD AN OVERHEAT EICAS WARNING FOR #2 ENG IN CARIBBEAN OCEANIC AIRSPACE.

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R ENG OVERHEAT EICAS MESSAGE. RAN CHKLIST NO R ENG OVERHEAT LIGHT. APPROX 5 MINS LATER; R ENG OVERHEAT LIGHT ILLUMINATED. WOULD NOT GO OUT WITH THROTTLE IN IDLE; SO WE SHUT DOWN R ENG IN ACCORDANCE WITH CHKLIST. AFTER CONSULTING WITH DISPATCH FOR DIVERT BASES (DISCUSSED GUATANOMO BAY; HAVANA; NASSAU; AND MIAMI) DECISION WAS MADE BY CAPT WITH INPUTS FROM BOTH FO'S THAT MIAMI WAS THE BEST CHOICE DUE TO POLITICAL PROBS IN CUBA; NASSAU CLOSED; AND MIAMI CLOSE BY. SINGLE ENG PROFILE; LNDG FLOWN BY CAPT TO UNEVENTFUL LNDG. SHOULD BE NOTED THAT ENG OVERHEAT LIGHT BULB WOULD ILLUMINATE WITH THE LIGHT TEST BUTTON DEPRESSED; BUT WOULD NOT ILLUMINATE PROPERLY WHEN A FIRE TEST WAS INITIATED. SUPPLEMENTAL INFO FROM ACN 510261: RAN R ENG OVERHEAT CHKLIST UP TO END SHUTDOWN POINT. GOT DISPATCH AND MAINT FOR ADVISE PRIOR TO SHUTTING ENG DOWN WITH JUST HAVING THE EICAS WITH NO OTHER INDICATIONS WAS UNUSUAL. LGT TEST TESTED GOOD. ENG FIRE TEST TESTED GOOD EXCEPT THE R ENG OVERHEAT LIGHT. AT THAT POINT; WE CONTINUED THE CHKLIST AND SHUT DOWN THE ENG; QRC DRIFTDOWN. HAVANA SAME DISTANCE AS MIA; GITMO BEHIND US PLUS BOTH FO'S WERE FAMILIAR WITH GITMO'S APCHS AND FOUND THEM A LITTLE TRICKY TO LAND A SINGLE ENG HVY. DISCUSSED NASSAU BUT IT WAS CLOSED. SUPPLEMENTAL INFO FROM ACN 510258: WE DECIDED TO GO TO MIAMI BECAUSE OF ITS WX. WE WERE ALL FAMILIAR WITH THE FIELD AND THE NAVAIDS. BOTH OF THE FO'S DID AN OUTSTANDING JOB WORKING THE PROBS; MAKING TIMELY AND ACCURATE SUGGESTIONS; AND EASING MY WORKLOAD. 20 MINS PRIOR TO LNDG; I ADVISED THE PURSER THAT A CABIN ADVISORY WAS TO BE PERFORMED AND TO LET US KNOW WHEN THE CABIN WAS READY. WE ALSO MADE AN ANNOUNCEMENT TO THE PAX AS TO THE PROB AND OUR LNDG INTENTIONS.