DA20 CARGO ACFT HAD ENG FLAMEOUT JUST PRIOR TO LEVELOFF. FLC REQUESTED LOWER ALT AND RESTARTED THE ENG. RPTR FO FAULTS CAPT AND MAINT FOR NOT WRITING UP THE MALFUNCTION AND CONTINUING TO FLY THE ACFT.

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CLBING TO CRUISE (FL350) AT FL349 ENCOUNTERED LIGHT TURB JUST PRIOR TO LEVELOFF. ENG FLAMED OUT AT FL349. CAPT IDENTED; PNF FO VERIFIED ENG. FO BEGAN CALLING CHKLIST FOR SECURING ENG. CAPT PF SECURED ENG. CAPT STATED NEED FOR LOWER ALT -- REQUESTED. FO LET HIM MAKE ATC CALL. CAPT REQUESTED LOWER; INFORMING ATC THIS WAS DUE TO PRESSURIZATION PROB. AT FL280 AIR START CHKLIST WAS INITIATED AND ENG STARTED. ALL SYS NORMAL. ONCE ALL CHKLISTS WERE COMPLETED; I ADVISED CAPT THAT WE NEEDED TO BE CERTAIN THAT NECESSARY RPTS; MISRS AND MIRRS WERE COMPLETED. CAPT AGREED AND THANKED ME FOR REMINDING HIM OF THIS. WE DEPLANED IN OKC; REVIEWING THAT THE CAPT WOULD MAKE APPROPRIATE CALLS. AFTER SOME TIME AND A COUPLE OF CALLS; CAPT SAID HE WAS AWAITING FURTHER INFO. CAPT RE-EMERGED TO STATE THAT THE DIRECTOR OF MAINT FELT THAT NO WRITE-UPS WERE REQUIRED AS THE ACFT HAD FLAMED OUT DUE TO 'ATMOSPHERIC CONDITIONS;' NOTHING MECHANICAL. WE WERE FREE TO CONTINUE THE TRIP. I INQUIRED AGAIN WHETHER A RPT WAS REQUIRED AND WAS TOLD THE DOM SAID NO. WE CONTINUED ON TO MMHO WITH NO WRITE-UPS AND UNBEKNOWNST TO ME; WITHOUT CONTACTING FLT OPS TO RPT THE INCIDENT. I ASSUMED THAT WITH SO MANY CALLS MADE; OPS MUST HAVE BEEN NOTIFIED -- IT WAS NOT. IN REVIEW; MY PROMPTING CAPT ABOUT RPTS WAS INSUFFICIENT. DUE TO LATENESS OF HR; HE BECAME FIXATED; I BELIEVE; ON THESE 2 ITEMS (AND WRITING UP FAILURE IN LOGBOOK). NOTIFYING FLT OPS WAS LEFT BY THE WAYSIDE. OPS CERTAINLY WOULD HAVE POINTED OUT OUR FAILURE TO WRITE UP ACFT AND GND IT APPROPRIATELY. THE DOM AND CAPT WERE INCORRECTLY FOCUSED ON ENG SYS OP AND NOT REG 135/121 FLT OPS REQUIREMENTS. I NEEDED TO BE MORE AGGRESSIVE AND CERTAIN THAT THE BIG PICTURE HAD BEEN ADDRESSED; AND NOT MERELY THE MECHANICAL ASPECTS. WE WERE NOT; AFTER ALL; MECHS; BUT PLTS. WE SHOULD NOT HAVE CONTINUED THIS FLT.