What makes them use GE or RR for their B777?
The 777-300ER/200LR are only available with the GE90 engine. In fact, for all practical purposes, the Trent 777 is out of production - Boeing hasn't delivered a Trent powered 777 in years. That's unlikely to change in the future, and the Trent isn't even an offering on the 777X.
As for SkiingPilot, the GE90 on the 777-300ER is currently rated at ~115,000 lbs. thrust. Please tell me which Rolls engine is currently rated at over 100,000 lbs. of thrust?