These are a set of instructions that worked for me to build CryEngine on Linux so I thought I'd share them.
NOTE: After 5.2.2 you can still build for Linux but the results are a black screen. As of 5.4 Linux versions do not even build. Crytek clearly has no focus on Linux and quite frankly unless you are a developer wanting to help develop CryEngine for Linux it is not worth your time to try and get things working on Linux. Even with 5.2.2 the airfield demo runs but has rendering issues. I would recommend looking at Unity and Unreal which both run on Linux.