How can I place my Exchange Server on a network with private addresses and have my MX record point to the server even though the server is behind a firewall?
Depends a little on the firewall, but in most cases you can set the MX record to point at the firewall and have the firewall forward the proper ports (Ports 110 and 995 at the least in most cases) to the Exchange server. The Exchange server will also need to be able to send mail out on Port 25 through the firewall.
The tricky part is that your Exchange server has a private address. If it had an Internet address but sat behind the firewall, you could just open the appropriate ports on the firewall to let the mail pass. You'd want to consult with your firewall expert, though, on the specific steps and the possible security consequences.