Since there is certainly more than one way for your program  to exit, maybe you
should separate the logic of you ExitAction class  to another method and/or
class. Then your ExitAction consists merely of a method call. The
windowlistener can then call that same method in response to a window closing
event. I am unsure how one would fire an event themselves so that ExitAction
could catch it. Perhaps someone else knows how to do this?

Implement a windowClosed() listener and call the ExitAction

You can use JFrame.addWindowListener(java.awt.event.WindowAdapter)

Maybe try something like this in your JFrame constructor:

setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
addWindowListener( new WindowAdapter()
{
public void windowClosing(WindowEvent e)
{
//TODO: - Call your Exit Action Code from here
}
} );

( or else use your JFrame object to make these calls.... )