I'm not sure what happened but the latest Java 7 seems almost useless for many websites and software. I've been forced to revert to Java 6 Update 45 so I can use many of my server/network management tools. I've noticed a number of websites also don't like the new Java, with chess.com's computer chess being one of them.
[edit: Newer installations of Java don't uninstall the older versions. You can try uninstalling Java 7 to see if that helps. If you don't have an older version of Java you can get it here: http://www.oracle.com/technetwork/java/archive-139210.html Just make sure to get the right version for your OS.]
I have to say, you computer chess doesn't work for me. I try to download java, follow all the direction, try the trouble shooting tips, and eventually have to give up when I have to go to work. Now the plugin doesn't even work on the Java website. So it isn't your fault. However, rather than being an easy quick install it is a time consuming effort I haven't even resolved yet. I don't have this problem on other sites so maybe you might follow a different approach.
[Yes, before a bunch of peope pipe in, I'm sure the problem is solvable. However, the fact remains that, unlike most things, I ran into a obscure problems on what should have been an easy install.)