Do ETOPS requirements vary between passenger, cargo, and non-revenue flights?
The ETOPS requirement is pretty much the same for all aircraft irrespective of usage as long as its under parts 121 or 135, as noted in the FAA InFO 07004, New ETOPS Regulations:


  The new ETOPS rules are now applicable to: 
  
  Part 121 
  
  • All two-engine airplanes more than 60 minutes flying time from an adequate airport (at a one-engine-inoperative cruise speed under standard conditions in still air). 
  
  • All passenger-carrying airplanes with more than two engines and more than 
  180 minutes flying time from an adequate airport (at a one-engine-inoperative 
  cruise speed under standard conditions in still air). 
  
  Part 135 
  
  • All multi-engine turbine-engine powered airplanes (other than all-cargo 
  operations with airplanes having more than two engines) more than 180 
  minutes flying time (limited to 240 minutes flying time) from an airport 
  meeting the requirements of part 135, §§ 135.385 or 135.393 and 135.219 at 
  an approved one-engine-inoperative cruise speed under standard conditions in 
  still air. 


As far as differences are concerned, ETOPS is waived only in one case:


  The FAA ... however agree that for airplanes with more than two engines, passenger carrying operations may be excluded from the ETOPS maintenance program requirements and that all-cargo operations may be excluded from all ETOPS requirements. 


The definition of ETOPS in the same document covers all aircraft except four engined all-cargo aircraft:


  Extended Operations (ETOPS) means an airplane flight operation other than an all-cargo operation in an airplane with more than two engines during which a portion of the flight is conducted beyond a time threshold identified in part 121 or part 135 of this chapter that is determined using an approved one-engine-inoperative cruise speed under standard atmospheric conditions in still air.


As far as non-revenue flights are considered, they are usually operated under Part 91, so the ETOPS rules don't apply. However, it is better to plan an ETOPS route for safety considerations (and to keep insurers happy).