Listeners apply to components or containers, not to areas within them. You'll need to store the coordinates of the rectangle you're interested in, and -in the listener- check whether the X/Y values of the event are within the rectangle. The Rectangle.contains method is useful for that.
You'll need a MouseMotionListener and listen to the mouseMoved event. Take care though, because there will be many of these events if you move your mouse across the component.
And speaking of the component, are you sure you want to override paint? With JComponent and its (direct and indirect) subclasses, it's recommended to override paintComponent. And the first call should be super.paintComponent(g);. You're missing the equivalent call in your method; if you still will be overriding paint, at least call super.paint(g) at the start of the method.
If you just have a Rectangle, then create a JPanel and add a LineBorder to it. Then add your "rectangle panel" to your GUI. Now you can add a MouseListener to the panel component and listen for MouseEvents.