Can the need for an engine overhaul affect the ETOPS rating?
Every time an engine is shut down in flight, it is tracked. 

Engine reliability does affect ETOPS certification. In addition to the aircraft being certified, each airline is also ETOPS certified. An aircraft or an airline with a poor record could have its ETOPS certification downgraded, or even potentially revoked. 

Wikipedia ETOPS Approval


  ETOPS approval is a two-step process. First, the airframe and engine combination must satisfy the basic ETOPS requirements during its type certification. This is called "ETOPS type approval". Such tests may include shutting down an engine and flying the remaining engine during the complete diversion time. Often such tests are performed in the middle of the ocean. It must be demonstrated that, during the diversion flight, the flight crew is not unduly burdened by extra workload due to the lost engine and that the probability of the remaining engine failing is extremely remote. For example, if an aircraft is rated for ETOPS-180, it means that it should be able to fly with full load and just one engine for 3 hours.
  
  Second, an operator who conducts ETOPS flights must satisfy their own
  country's aviation regulators about their ability to conduct ETOPS
  flights. This is called "ETOPS operational certification" and involves
  compliance with additional special engineering and flight crew
  procedures in addition to the normal engineering and flight
  procedures. Pilots and engineering staff must be qualified and trained
  for ETOPS. An airline with extensive experience operating long
  distance flights may be awarded ETOPS operational approval
  immediately, others may need to demonstrate ability through a series
  of ETOPS proving flights.
  
  Regulators closely watch the ETOPS performance of both type
  certificate holders and their affiliated airlines. Any technical
  incidents during an ETOPS flight must be recorded. From the data
  collected, the reliability of the particular airframe-engine
  combination is measured and statistics published. The figures must be
  within limits of type certifications. Of course, the figures required
  for ETOPS-180 will always be more stringent than ETOPS-120.
  Unsatisfactory figures would lead to a downgrade, or worse, suspension
  of ETOPS capabilities either for the type certificate holder or the
  airline.