Adding a JButton to a Canvas

Unless you're doing some super low level work, you should avoid using Canvas, in your case, a JPanel will do just fine.

If you override a method of a class, you should beware of what that method does and either be prepared to replicate its work (as much as your implementation requires it to) or call it's super method.

I would recommend you start by looking at Painting in AWT and Swing and Performing Custom Painting to get a better understand of how painting works in Swing.

I would also encourage you to decouple the workflow, separating the various aspects of your system, to the point you have a "game board" component, which, all it does it what ever is needed of the game board and a seperate component for dealing with things like, "try again".

If you're clever, you can easily overlay this panels on top of each other and even do some neat effects with them

This is an extremely overly simplified example, but it's intention is to demonstrate:

1. Decoupling of responsibility
2. The utilisation of the "observer pattern" to achieve point 1

import java.awt.Color;
import java.awt.Dimension;
import java.awt.EventQueue;
import java.awt.Graphics;
import java.awt.Graphics2D;
import java.awt.GridBagConstraints;
import java.awt.GridBagLayout;
import java.awt.RadialGradientPaint;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JPanel;

public class Test {

    public static void main(String[] args) {
        new Test();
    }

    public Test() {
        EventQueue.invokeLater(new Runnable() {
            @Override
            public void run() {
                JFrame frame = new JFrame("Chess");
                JPanel contentPane = new JPanel(new GridBagLayout());
                GridBagConstraints gbc = new GridBagConstraints();
                gbc.gridx = 0;
                gbc.gridy = 0;
                gbc.weightx = 1;
                gbc.weighty = 1;
                gbc.fill = gbc.BOTH;

                GameListener listener = new GameListener() {
                    private AlertPane alertPane;

                    private AlertPane getAlertPane() {
                        if (alertPane != null) {
                            return alertPane;
                        }

                        alertPane = new AlertPane();
                        alertPane.setGameListener(this);
                        return alertPane;
                    }

                    @Override
                    public void gameWasCompleted() {
                        contentPane.add(getAlertPane(), gbc);
                        contentPane.setComponentZOrder(getAlertPane(), 0);
                        contentPane.revalidate();
                        contentPane.repaint();
                    }

                    @Override
                    public void startNewGame() {
                        contentPane.remove(getAlertPane());
                        contentPane.revalidate();
                        contentPane.repaint();
                    }
                };

                MainPane mainPane = new MainPane();
                mainPane.setGameListener(listener);

                contentPane.add(mainPane, gbc);

                frame.setContentPane(contentPane);
                frame.pack();
                frame.setLocationRelativeTo(null);
                frame.setVisible(true);
            }
        });
    }

    public interface GameListener {

        public void gameWasCompleted();

        public void startNewGame();
    }

    public class AlertPane extends JPanel {

        private GameListener gameListener;

        private JButton try144Button;

        public AlertPane() {
            setLayout(new GridBagLayout());
            setOpaque(false);

            try144Button = new JButton();
            try144Button.setText("CLICK ME");
            try144Button.setBounds(210, 60, 150, 150);

            add(try144Button);

            try144Button.addActionListener(new ActionListener() {
                @Override
                public void actionPerformed(ActionEvent e) {
                    gameListener.startNewGame();
                }
            });
        }

        public void setGameListener(GameListener gameListener) {
            this.gameListener = gameListener;
        }

        @Override
        protected void paintComponent(Graphics g) {
            super.paintComponent(g);

            Graphics2D g2d = (Graphics2D) g.create();
            Color startColor = new Color(255, 255, 0, 0);
            Color endColor = new Color(255, 255, 0, 192);

            RadialGradientPaint rgp = new RadialGradientPaint(
                    getWidth() / 2, getHeight() / 2,
                    Math.max(getWidth(), getHeight()),
                    new float[]{0f, 0.25f},
                    new Color[]{startColor, endColor});
            g2d.setPaint(rgp);
            g2d.fillRect(0, 0, getWidth(), getHeight());
            g2d.dispose();
        }

    }

    public class MainPane extends JPanel implements MouseListener {

        private GameListener gameListener;

        public MainPane() {
            addMouseListener(this);
        }

        public void setGameListener(GameListener gameListener) {
            this.gameListener = gameListener;
        }

        @Override
        public Dimension getPreferredSize() {
            return new Dimension(1000, 1000);
        }

        @Override
        protected void paintComponent(Graphics g) {
            super.paintComponent(g);
            int xshift;
            int yshift = 0;

            g.setColor(new Color(0, 0, 0));
            for (int i = 0; i < 4; i++) {
                xshift = 125;
                for (int j = 0; j < 4; j++) {
                    g.fillRect(xshift, yshift, 125, 125);
                    xshift += 250;
                }
                xshift = 0;
                yshift += 125;

                for (int k = 0; k < 4; k++) {
                    g.fillRect(xshift, yshift, 125, 125);
                    xshift += 250;
                }
                yshift += 125;
            }
            initalprnt(g);
        }

        protected void initalprnt(Graphics g) {
            int xshift = 125;
            int yshift = 0;
            g.setColor(new Color(0, 100, 0));
            for (int i = 0; i < 3; i++) {
                if (i == 2) {
                    xshift = 125;
                }
                for (int j = 0; j < 4; j++) {
                    g.fillOval(xshift, yshift, 125, 125);
                    xshift += 250;
                }
                xshift = 0;
                yshift += 125;
            }
            yshift += 250;
            g.setColor(new Color(0, 0, 100));
            for (int k = 0; k < 3; k++) {
                if (k == 2) {
                    xshift = 0;
                }
                for (int j = 0; j < 4; j++) {
                    g.fillOval(xshift, yshift, 125, 125);
                    xshift += 250;
                }
                xshift = 125;
                yshift += 125;
            }
        }

        public void mouseClicked(MouseEvent e) {
            gameListener.gameWasCompleted();
        }

        public void mousePressed(MouseEvent e) {
        }

        public void mouseReleased(MouseEvent e) {
        }

        public void mouseEntered(MouseEvent e) {
        }

        public void mouseExited(MouseEvent e) {
        }
    }
}