Virtual Files can create per character copies (or in some cases per team copies) which get saved to the same place that the game saves the usual files. So the answer is, wherever the game normally saves files is where the virtual files will be.
As for not saving. It is up to the game to save files. ISBoxer doesn't do that. If you have a virtual file, then that is just a redirected file name, so the game says I want to access file document.txt, and ISBoxer intercepts and says sure, you can use document-virtual1.txt instead. The game still reads normally, and writes normally, but just uses the redirected file name.
If you don't have a virtual file, it is still up to the game to save it.
There have been the odd occasions where game updates have changed how they access files, which has then resulted in some needed changes to the virtual files feature, but so far, this is the first report of this issue with EVE, so I suspect it is probably more something local.
The default virtual files for EVE don't include the settings folder anymore, which some people like to use. What Virtual files do you have? Maybe the issue is elsewhere and you get an error when closing the game that you're not telling us about? Perhaps a
compatibility diagnostic will show up something that might be causing an issue.