Engine problems are hardly the only Boeing issue in recent years. Nor am I inclined to give them a pass on those; they are most certainly heavily involved in the engine design and production process, as they are on most outsourced parts they use.
The 777 was fitted with at least three different engines, from General Electric, Pratt & Whitney, or Rolls-Royce. Customer airline normally selects which engine they prefer. Some of these same engines are used on competing aircraft from Airbus.
The 777-200 is a 1994 design, way before all the recent trouble started. Its three engine types have been produced thousands of times, the GE90 has an excellent in flight shutdown rate of one per million flight hours for example.