Support for the 777-300 is there with the latest GSX Live Update.
IMPORTANT: Remove any GSX.CFG you have in your %APPDATA%\Virtuali\Airplanes\PMDG 777-300ER folder, if you added it before this update.
You can further customize the airplane using the GSX editor, but it's important you remove any unofficial GSX.CFG file you might have had already, otherwise it will take precedence on the internal GSX data, which adds support to check the additional ground connections (air start, gpu, chocks, etc.).