Well, I can get it to work, but I don't really like the process...
Change the squid line to "su -c '/sbin/service squid restart'".
Then, right-click the desktop link you are using to run the script, and choose Properties, Application, Advanced Options. Set the link to run in a terminal.
Now, when you run the link, it will open a terminal, and ask for the root password. If you give it the correct password, it will run restart squid. If you give it the wrong password, it won't. Either way, it will then start Mozilla.
The part I don't like is that if somebody can get into your script, they can put something malicious in the su command, and you would run it not knowing. I would just open a terminal and type in the su command when I needed to.