OVERVIEW:
I am running a script on one machine that connects to a MySQL database on another machine that is outside of our university's domain. According to the administrator, network policies do not allow the compute nodes to access machines outside of our university's domain.

COMPUTERS:
A = compute node within university
B = 2nd machine within university that does not block outside connections
C = machine outside of university
mysqldb on A cannot connect to C ....but.....
mysqldb on A can connect to B

WHAT I WOULD LIKE TO DO:
Is it possible to set something up where A talks to a port on B, and that port is actually nothing more than 3306 on C?

This belongs in the Hardware/Software -> Networking forum... If you know how to use ssh to create tunnels go that route. If you don't know how to do that, search google for the numerous guides that exist.