This week's book giveaway is in the OCAJP forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide 1Z0-808 and have Jeanne Boyarsky & Scott Selikoff on-line! See this thread for details.
I don't know of a way to handle the back button specifically, but if you want to handle the generic event of the user leaving your page, you can override the stop() method of the applet. This method will be called whenever the user navigates away from the page that spawned the applet.