I'm not military but have Single, Multi and Instrument ratings.
The definition of 'critical engine' is the engine which, if failed, will most adversely effect the flying of the airplane.
This is important on piston twins and turboprops (not so much for reasons we will get into later) that have conventional engines which means both props spin the same way. If the engines have counter-rotating props (P-38, Piper Seneca) then one engine is not worse than the other, thus no critical engine.
The reason behind the critical engine is because the descending blade creates more thrust than the ascending blade and that means the arm between the Center of Gravity and the point of max thrust is further on the right engine than the left engine (we're talking conventional twins, where the prop spins clockwise when viewed from the cockpit) because the descending blade on the right engine is farther away from the CG than that of the left. That makes the left the critical engine because if it fails, then the right engine can really whip that airplane around and it is harder to fly than if the left engine fails.
Turbine engines don't have this problem because there is no windmilling propeller. Turboprops such as a King Air 350 and damn near every other one these days have an 'auto feather' which if an engine fails, all the oil is forced out of the hub and the prop feathers automatically without any input from the pilot...this is handy because if you lose an engine above V1, you want to continue the takeoff.
Turbine engines also don't have any problems with torque, in fact some jet engines have compressor fans that rotate opposite one another.
By the way...where do you go to school so that you can see MiG's taking off...I wanna go there