I don't think I've seen any developer using EAC outright poo-poo the possibility so far, so that's somewhat encouraging...
None of the Kernel anti cheat work without being compatible with Linux which currently none of them are. EAC is supposed to be in the future but I doubt it will work.
It's not impossible, it's just up to Activision.
[1] https://dev.epicgames.com/en-US/news/epic-online-services-la...
[2] https://twitter.com/thebattleye/status/1441477816311291906
Unless out of spite, like when Snapchat banned third-party clients for Windows Phone (like 6snap) and accounts using it (I'm still raging that my 'benbristow' username got banned and now I'm using 'benbristow1' because of that).
Fun fact: The guy that made 6snap (and other 6 prefixed apps like 6tin (Tinder)) (Rudy Huyn) was hired by Microsoft and is now one of the leads behind the updated Windows 11 Store app
No, it wont.