I wont get into "what you need to do at job place" - the "contact your sysadmin to ask him about" was already stated.
Next, have a look at GNU httptunnel (actually you can install it from cygwin since it's in the package list) and you may even tunnel the ssh connection. The drawback - you need it both running on the server end and client end so you'll have to ask your shell provider to run it too (which I doubt they'll do) or... set up a httptunnel redirector at home computer.
The other possible "tunneling" software might be corckscrew - have a look here and on its homepage.
Final note - it might be easier to reason with your sysadmin, use a live-cd or vmware-install to get around with linux instead of wasting energy into 'getting around' - an aware sysadmin will get you. So, if you want to do the shell IRC-ing, do it from home and not at work.