Okay, so I made sure that my Java was more up-to-date than the version number required by the page; I updated it further, anyway.
And, the Java applet in the greyed-out window still did not work; in either Chromium or Firefox.
Same for me, but then I stopped and restarted firefox (iceweasel actually) and the java applets then worked.
I'm using a Debian squeeze (stable) system, had to install jdk-7u3-linux-x64.tar.gz from oracle using these instructions:
and got it working eventually.
(Thanks crazysquirrel for sane instructions for Debian systems.)