Need Multiple JPanel's to respond its own mouse click events.

Answer 1

The problem would appear to be that you have a class variable and a local variable for each of your square panels.

Square square1 = new Square("1"); 

should be:

square1 = new Square("1"); 

now you will only have a class variable which the PuzzleListener can reference.

Answer 2

Is there a need to use a single MouseListener? You could create a new instance of your PuzzleListener and add it to each JPanel. That way there is no confusion as to which listener on which panel is getting fired.