It took a while, but I've been able to breach the proxy, finally. All it took was some disguising (via SSL encryption) of the outgoing packets, sending the packets through the HTTPS port, and figuring out a way to do NTLM authentication manually. With all that being said, it took a bunch of tools, some setup, and whole lot of reading. The end result is a successful encrypted connection to an external SSH server.
I plan on posting a more detailed guide on how to do this in a later post as I currently have to run to make it to a prior commitment.