In Debian, I was able to get the WebEx java applet to launch in Iceweasel with the help of OpenJDK, but I couldn't get screensharing to work properly. It just never came up when someone was sharing their screen and it wouldn't let me share my own.
Go ahead and install some 32-bit libraries, this should help:
Note: You must have multiarch enabled on your machine for this to work:
dpkg --add-architecture i386; sudo apt-get update
sudo apt-get install libgtk2.0-0:i386 libglib2.0-0:i386 libglib2.0-0:i386 openjdk-7-jre-headless:i386 libpango-1.0-0:i386 libpangoft2-1.0-0:i386 libpangox-1.0-0:i386 libxft2:i386 libxmu6:i386 libxt6:i386 libxv1:i386 openjdk-7-jre
Then launch your meeting again. Hopefully screensharing should work. I have heard reports of this not helping with meeting audio issues, but I don't use WebEx audio, I can't comment on that.
If that doesn't work, figure out what else you're missing with the help of this AskUbuntu post.
In case that post ever goes away, I've mirrored the steps here:
From this post, here is a step-by-step method that might work:
ldd $HOME/.webex/????/*.so > $HOME/check.txt
grep "not found" $HOME/check.txt | sort | uniq
libasound.so.2 => not found libjawt.so => not found libXmu.so.6 => not found libXtst.so.6 => not found libXv.so.1 => not found
sudo apt-get install apt-file sudo apt-file update
apt-file search libXmu.so.6 apt-file search libjawt.so
sudo apt-get install -y libxmu6:i386 sudo apt-get install -y libasound2:i386 sudo apt-get install -y libxv1:i386 sudo apt-get install -y libxtst6:i386 sudo apt-get install -y libgcj12-awt:i386