The error I was getting looked like this:

coldfusion.jsp.CompilationFailedException: Errors reported by Java compiler: jikes: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory .

The fix is super simple, Just run this in a terminal:

sudo apt-get install libstdc++5

sudo apt-get install sun-java6-jre sun-java6-plugin sun-java6-fonts

Next I checked to make sure the upgrade went as planned by checking the installed Java version using the following terminal command:

java -version

However, it would seem the root of the problem was the Mozilla plugin directory did not exist. In the end I was able to fix the plugin for Firefox by recreating the plugins directory in my Home folder and creating a link to the Java 6 JRE libjavaplugin_oji.so using the following steps (again in terminal):

mkdir .mozilla/plugins
cd .mozilla/plugins
ln -s /usr/lib/jvm/java-6-sun/jre/plugin/i386/ns7/libjavaplugin_oji.so

This was all it took to get the Java plugin functioning again in Firefox 2.