Personally, I actually was able to get EGS to install on CrossOver, however, that was with an older version of EGS, and an older version of CrossOver, this was around 2021, and DXVK have to be disabled to get the UI to show up correctly
I had also personally even able to get Fortnite to run on my Linux from EGS, even though it was for 10 minutes only and gameplay was very laggy, but the fact that I was able to get it to run at all in CrossOver, is impressive.
Get that bottle created, and start the following step
Install all of the .NET and Visual C++ Redistributable from the CrossTie (Install Software Into Bottle), and "DirectX for Modern Games", into your EGS bottle. I personally made a "DefaultTemplate" bottle with all of these softwares included, and all I need to do is duplicate the bottle.
Also make sure to disable DXVK in CrossOver to get the UI to display correctly. You can enable it again when you want to launch a game.
Then please download the 1.2.17.0 version of EGS from here
https://downzen.com/en/windows/epic-games-launcher/download/12170/
For some reason, if you download the latest version (16.2.1 at the time of this post), it crashes on start up, but if an older version is installed, it runs fine. Rest assured that you do not need the latest EGS for your games library, there are very little improvements between the two versions.
Once the above version is installed,. it will start an update around 500mb, you can either let this update go through or close it, it won't matter either way because you will be getting the "Self-Update Failed" message.
This is when you will be adding the "-SkipBuildPatchPrereq" in the the Epic Games shortcut (without the quotes), to do this on Linux, right click the EGS icon, Properties, Application Tab, then add the code to Argument field with a space, this will tell EGS to not update itself.
Then start EGS again, it will somewhat shows that an update is installing, but it is really not, then you should be getting a message saying "An update is required, continue?" You will have a button saying "continue", click on it and another update will start. HOWEVER, this update will only be about 331mb, let this one go through, and then you will be presented with the UI.
Additional troubleshooting:
If EGS still crashes on boot, go in CrossOver and click on "Quit All Applications", this will simulate a Windows shut down and reboot, and once it says "Ready" again on the top right, try opening EGS again.
Chances are, EGS will open and close quite a few times, and it will seem to be stuck in an app start loop, just give it a few minutes, the app will eventually stay open and display the UI, and as mentioned above, make sure DXVK is disabled when attempting to open EGS.
Reason why I mentioned "Hit and Miss" in the title was because that even with these steps, EGS only wants to work sometimes, most of the time it just flat out crashes with something like "CrashReportClient.exe" or "EpicWebHelper.exe". With all that time spent trying to get this to work, you might as well just download Heroic Games Launcher and figure out a way to get the games downloaded to use CrossOver.
Proof:
https://imgur.com/a/1Vg5cLz