I am a student of Japanese and I was wondering how I would go about
chatting on Yahoo! JAPAN's chat rooms.

I have WindowsXP with Asian fonts installed as well as the MS Japanese
Input Editor. I'm using Mozilla using the Java 1.4.2_05 JRE. I can
access chat rooms and the Java Applet loads fine, but all characters
appear as square blocks. I am able to see what characters I input in
the send line, but when I hit enter, they too appear as empty blocks
in the chat room.

My question is simply how can I fix this so I can read all the
characters normally in the Applet. Do I need to install a different
version of the JRE or do I need to configure something differently?

y not using internet explorer

Check this out, these guys are working on a Japanese Mozilla project
that may help you out.

http://www.mozilla.gr.jp/wazilla/index-en.html

I choose not to use Internet Explorer because it is a terrible piece of software.
On top of that, I tried using IE to get the chat to work, but it did
not load the applet correctly.

Hello,

I'm not sure if this will solve your problem or not, but it is worth a try. 

1) Go to start, open the control panel.
2) Find "Regional and Language Options" and open it.
3) Click on the "Advanced" tab.
4) Under the "Language for non-unicode programs" pick "Japanese"
instead of English.
5) Probably, reboot your computer.

Be warned that this will make a fair number of programs look different
and will change your backslash character into a yen sign. However, if
you don't do this, some japanese programs won't work on your computer
(notably, iTunes won't play files with japanese names, among others).

Hope this helps!