B767-300 DIVERTS ENROUTE TO ZZZZ DUE TO INSUFFICIENT FUEL AT ARRIVAL INDICATIONS. DISCOVER EXTENSIVE DAMAGE TO RIGHT ENGINE TAIL CONE/SHROUD.

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PUSHBACK AND ENG START NORMAL; IMMEDIATELY FOLLOWED BY ANTI-ICE ON AND AN ENG RUN-UP. TAXI NORMAL. TKOF POS ONE MORE ENG RUN-UP AFTER A LESS THAN 10 MIN TAXI. CAPT'S TKOF NORMAL UNTIL AIRBORNE; THEN A SLIGHT PULL TO THE R COUNTERED WITH L RUDDER TRIM (FAIRLY NORMAL). PASSING 4000 FT TO AROUND 12000 FT ERRATIC L ENG FUEL FLOW; GOING DOWN TO AS LOW AS 3000 LBS PER HR BACK UP TO A MATCHING INDICATION WITH THE R FUEL FLOW. ALL OTHER INDICATIONS NORMAL. FO #2 INFORMED MAINT CTL. FO #1 THOUGHT HE FELT A SLIGHT VIBRATION DURING CLB. WE CHKED ALL DRAG DEVICES; PLUS APU AND FOUND THEY WERE ALL SECURED PROPERLY. VIBRATIONS THOUGHT TO POSSIBLY BE LIGHT TURB. CAPT STATED THAT IT SEEMS TO BE A SLOW CLBOUT; BUT HE HAD NOT FLOWN AN AIRPLANE THIS HVY (390000 LBS PLUS) IN A WHILE. MAINT CTL REPLIED THAT FUEL QUANTITY INDICATIONS MAY BE IN ERROR DUE TO THIS GAUGE. CRUISING FL320-330 EARLY ON FUEL QUANTITY A BIT LOW; BUT 'PROG' PAGE LNDG FUEL NORMAL. MID-FLT STEP CLB TO FL350 RESULTED IN FUEL FLOWN GOING UP DRAMATICALLY AND LNDG FUEL DROPPING STEADILY. FUEL LEAK CHKLIST RUN; TANKS EVEN; JUST A HIGH BURN RATE. CONTACTED DISPATCH WITH PHONE PATCH TO MAINT CTL. WE ALL DETERMINED THE BEST COURSE OF ACTION WAS TO DIVERT TO ZZZZ. ASKED FOR AND RECEIVED LNDG DATA FROM DISPATCH; COORDINATED DIVERT WITH ATC. ADVISED FLT ATTENDANTS TO DIVERT AND REASON WHY. INFORMED PAX WE NEEDED MORE FUEL TO MAKE TO DEST ARPT SO WE WERE GOING TO LAND AT ZZZZ. DAY VFR LNDG FLAPS 30 DEGS; AUTOBRAKES 3; ALL NORMAL UNTIL ATTEMPTED REVERSE THRUST AND THE R REVERSER WOULD NOT GO INTO REVERSE. PLENTY OF STOPPING DISTANCE EVEN WITHOUT REVERSE. NORMAL TAXI-IN AND ENG SHUTDOWN. WALKAROUND REVEALED AFT R ENG SHROUD SEVERELY DAMAGED WITH PARTS OF IT MISSING. ADVISED DISPATCH TO CONTACT THE DEP ARPT TO CHK FOR PIECES AND PARTS ON THE RWYS SO THAT NO OTHER ACFT WERE DAMAGED. NO PIECES OR PARTS FOUND. SUSPECT ENG SHROUD CAME UNLATCHED DURING ANTI-ICING RUN-UPS PRIOR TO TKOF; THEN PIECES AND PARTS RIPPED FREE DURING STEP CLB.