Java, replacement for infinite loops?

I would suggest using a seperate thread to handle the array. Make sure you are using thread safe object (check Java Docs) and simply call .start() on your thread object when you want to start. Keep a pointer to it so you can pause it via setPaused(true)

Something like this....

class MyArrayUpdater extends Thread {
    private volatile boolean enabled = true;
    private volatile boolean paused = true;

    @Override
    public void run() {
        super.run();
        try {
            while (enabled) {
                if (!paused) {
                    // Do stuff to your array here.....
                }
                Thread.sleep(1);
            }
        } catch (InterruptedException ex) {
            // handle ex
        }
    }

    public void setEnabled(boolean arg) {
        enabled = arg;
    }

    public void setPaused(boolean arg) {
        paused = arg;
    }
}

What you need is use a Timer which changes the state of your component ( in this case the cellular growth ) and then call JComponent.repaint()

This timer can be cancelled to make the pause and then to restart it, you just create a new one:

So you could define the following two methods:

private Timer timer;
...
public void startPaiting() {
    timer = new Timer();
    timer.schedule( new TimerTask(){
        public void run(){
            changeState();
            repaint();
        }
    },0,  10000 ); // 10 s. 
}

public void pause(){
    timer.cancel();
}

And then in your "Pause/Resume" button invoke this "pause/startPaiting" methods:

if( e.getActionCommand().equals("Pause")){
    growPanel.pause();
    setText("Resume");
} else {
    growPanel.startPaiting();
    setText("Pause");
}

Here's the complete source code to see it running:

import javax.swing.*;
import java.awt.Graphics;
import java.awt.Point;
import java.awt.event.*;
import java.util.Timer;
import java.util.TimerTask;

public class Grow {

    public static void main( String [] args ) {
        JFrame frame = new JFrame();
        final GrowPanel growPanel = new GrowPanel();
        frame.add( growPanel );
        frame.add( new JPanel(){{
            add( new JButton("Pause"){{
                addActionListener( new ActionListener(){
                    public void actionPerformed( ActionEvent e ){
                        if( e.getActionCommand().equals("Pause")){
                            growPanel.pause();
                            setText("Resume");
                        } else {
                            growPanel.startPaiting();
                            setText("Pause");
                        }
                    }
                });
        }});}}, java.awt.BorderLayout.SOUTH );
        frame.setSize( 400, 300 );
        frame.setVisible( true );
    }
}

class GrowPanel extends JComponent {
    private int x;
    private int y;
    private Timer timer;
    GrowPanel() {
        x = 10;
        y = 10;
        startPaiting();
    }

    public void startPaiting() {
        timer = new Timer();
        timer.schedule( new TimerTask(){
            public void run(){
                changeState();
                repaint();
            }
        },0,  100 ); // or 10000 which is 10 s. 
    }

    public void pause(){
        timer.cancel();
    }

    public void paintComponent( Graphics g ){
        g.fillOval( x, y, 10, 10 );
    }
    private void changeState(){
            x+=10;
            if( x >= 400 ) {
                y+=10;
                x = 0;
            }
            if( y >= 300 ){
                y = 10;
            }
    }

}

If those "buttons" are Swing buttons, then the way to do this is: have the Start button create a new javax.swing.Timer object which does the update every 10 seconds. Then have the Pause button stop that timer.