No. B777 can travel anywhere the range allows but their flying route must be supported by a suitable landing airport which has to be no more than 180mins away. B777 can still fly 8 hours LHR to JFK (london to new york) so long as there is an suitable landing strip within 180mins. Eg, if the Transatlantic flight of 4 hours has no other island, the B777 technically cannot fly this route cos at the 4 hour mark, the nearest airport (the destination) is 4 hours away. However, if there is an island with a suitable landing strip along the way (say at mid point) then no problem.
Nobody plans an engine failure. So even there is no engine failure on the B777, ETOPS applies. ETOPS is only in event of engine failure, but the aircraft must stick to ETOPS rules even if both engines are running smoothly