If you want the GUI to behave as if you clicked the X close button then you need to dispatch a window closing event to the Window. The ExitAction from Closing An Application allows you to add this functionality to a menu item or any component that uses Actions easily.
frame.dispatchEvent(new WindowEvent(frame, WindowEvent.WINDOW_CLOSING));
setVisible(false); //you can’t see me!
dispose(); //Destroy the JFrame object
Not too tricky.