наверное добавить лисенер на саму форму окна приложения, тогда событие будет обрабатываться во всех случаях, когда окно приложения активно.

package application;

public class Application {

    public static Snake s;

    public static void main(String[] args) {
        s = new Snake();
        s.go();
    }
}

package application;

import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.KeyAdapter;
import javax.swing.*;
import java.awt.event.KeyEvent;

public class Snake extends JComponent {

    int x = 350;
    int y = 350;
    public MyPan drawPanel;
    JFrame frame;
    Timer timer = new Timer(100, new MyKey());

    public void go() {
        frame = new JFrame();
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        drawPanel = new MyPan(this);
        timer.start();
        frame.getContentPane().add(drawPanel);
        frame.setSize(700, 700);
        frame.setVisible(true);

        frame.setFocusable(true);
        frame.setFocusTraversalKeysEnabled(false);
        frame.addKeyListener(new MyKey());
        frame.setFocusable(true);
    }

    public void moveRight() {
        x += 10;
        try {
            Thread.sleep(20);
        } catch (Exception e) {
            e.printStackTrace();
        }
    }

    public void moveLeft() {
        x -= 10;
        try {
            Thread.sleep(20);
        } catch (Exception e) {
            e.printStackTrace();
        }
    }

    public void moveDown() {
        y += 10;
        try {
            Thread.sleep(20);
        } catch (Exception e) {
            e.printStackTrace();
        }
    }

    public void moveUp() {
        y -= 10;
        try {
            Thread.sleep(20);
        } catch (Exception e) {
            e.printStackTrace();
        }
    }

    public class MyKey extends KeyAdapter implements ActionListener {

        @Override
        public void keyPressed(KeyEvent e) {
            if (e.getKeyCode() == KeyEvent.VK_RIGHT) {
                moveRight();
            }
            if (e.getKeyCode() == KeyEvent.VK_UP) {
                moveUp();
            }
            if (e.getKeyCode() == KeyEvent.VK_LEFT) {
                moveLeft();
            }
            if (e.getKeyCode() == KeyEvent.VK_DOWN) {
                moveDown();
            }
        }

        @Override
        public void actionPerformed(ActionEvent e) {
            drawPanel.repaint();
        }
    }
}

package application;

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

public class MyPan extends JPanel {
    //  Snake snake = new Snake();

    private Snake s;

    public MyPan(Snake ss) {
        s = ss;
    }

    public void paint(Graphics g) {
        g.setColor(Color.BLACK);
        g.fillRect(0, 0, 700, 700);
        g.setColor(Color.GREEN);
        g.fillRoundRect(s.x, s.y, 50, 50, 40, 40); 
    }
}