If you're still trying to use the same mail server, that's probably your problem. You're not longer on the local network at your school. So when you connect to the school server, it sees you as an outside connection and refuses the connection. Most well-secured mail servers will not allow an outside connection at all, or require authentication in order to send mail, or even require SSL encryption to work (AT&T Broadband now requires this, and uses non-standard ports for it). Most allow you to receive mail however even from off their network, since you can't really abuse that.
In the settings in Outlook for your mail account, there's an option for the outgoing mail server to log in when sending mail. Check that option and then try again. If that doesn't work, contact the mail administrators and ask them if they allow connections from off their network.
The fact that your connection is being shared through your Linux machine will not affect whether your mail client can send or receive mail.
