Is there a way to make Foobar2000 play .cue files? Currently on my system VLC Media Player plays them.
Did you try it?
In other words, yes.
However, note:
"Exclude file types" can be set to empty to allow all types. By default, it is set to "*.CUE" to ignore cuesheets when adding whole folders.
Just click on .cue with right mouse button, choose "Open with" and choose foobar2000 - proceed to "other apps" or search for exec when needed. You can also try go to "Manage file type associations" in Windows Control Panel. Foobar got such link/button in its configuration (Ctrl+P) in Shell Integration section.
foobar2000 will not play non-compliant CUEs, as generated by EAC.