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Quit = new QuitButton();

This is a bad idea. You've told Quit that it's an object of class QuitButton.

Unless you have a class called QuitButton.

Your whole class is a JPanel. Why are you adding another JPanel to it?

I've done it where I've had a whole class that was a JFrame but I used a JPanel to put everything on, that way I could add...

import javax.swing.*;
import java.awt.*;

@SuppressWarnings ("serial")

public class GameOfLifeApp extends JFrame {

// constructor
public GameOfLifeApp (String title, String fname) {
//...