Is the air in a jet engine supercritical?
Maybe at takeoff, definitely not in cruise

You have to consider that atmospheric pressure drops with altitude. At FL360 you have ~0.226 bars. Even multiplying with the GE9X's pressure ratio of 60, you get ~13.56 bars, well below the 34 bars needed.  

Takeoff might be a different situation, since the engines are operating at full power with ambient pressure ~1 bar, this could lead the high pressure compressor stages to operate in supercritical conditions. Note though that the combustion that immediately follows changes significantly the chemical composition, and thus the critical point might be altered as a consequence.

As engines are optimized for cruise conditions, it is highly unlikely that supercritical conditions have extensive design consequences, unless they would render takeoff impractical.