If they really are as computer illiterate as you say, just create a shortcut for Mozilla and put it in a folder that only you access, you can even hide the file (check the "hidden" box under properties) if you want. Then delete the shortcuts that are in plain site on the desktop and/or in the startup menu.
I have several programs that would let you clean it out, but...people may frown upon the methods you'd need to use in order to get them to work, so I won't.
Some things to see if there's a trial version of (I don't know if there is, these are just programs I have):
Spyware Doctor
Spybot Search and Destroy
Registry Mechanic
PcBug Doctor
avast! Antivirus
CCleaner
XoftSpySE
Also try downloading hijackthis, a free program that'll scan your computer and give you a log of practically everything. There are then loads of forums online with techs who will look at the log and give you a fix for clearing out any problems.
Also, those free programs you tried and didn't work, make sure you run them in safe mode. Also try searching google for the trojan itself, just type in the name of it. You should get several things on how to remove it.