I'm not sure what's causing this, I've run the installer multiple times and it keeps happening. Do I need to pick another server? The shortcut will not work otherwise.

I think the ezquake-gl.exe was recently removed/renamed to just ezquake.exe and probably the shortcut in the nQuake package was not adjusted with regards to this, i.e. a bug in the nQuake package. ezQuake.exe is actually ezQuake-gl.exe, just change the path in the shortcut link.

Edit: noticed the issue is already reported and even pull requested in the nQuake repo.

Also choose the FNU mirror or whatever it is...

This should now be fixed in the nQuake installer.

(Thanks to Meag for the pull request)