Made a mod like this myself a while back, but was lost in a server crash (and of course
I was too stupid to back the modifications up

). This one of course being much better, namely in features, so many thanks
@stivo20
Mouselike.org are just running their own irc server, so you are still connecting to them and whichever port their server is setup on directly. Off hand I don't recall if the CGI clients out there are able to work around firewalls or not.
If you have the ability however to setup your own irc server, you may open up other ports on it besides 6667. In this way, you can direct the webbased chat to connect to that port, rather than 6667.