Make the firewall administrator your best friend, and ask him to deny access to/from UDP 6667-6669 ports of public IP addresses, or turn off socks proxy (if mIRC goes that way).
Other options include uninstalling mIRC every day, placing a video camera behind the user, having a bodyguard watch him/her, etc ;)
Now, seriously, install NT and manage the permisions and rights of the users. Audit the actions they do, too. They won't be able to see mIRC anymore, except in computer magazines. Nor java chat applets for that matter.
Or if you're really angry at them install linux on their machines :)
For windows there's a program called netmonitor (at www.winfiles.com) that logs every connection on the computer it's running it. But then they might close it when they run mIRC.