BitCoin Miner - Remote Miners with SSH Access only I've graciously been given a big opportunity with a server infrastructure to work with, and I'd like to start it off by starting some Bitcoin mining, however there's a catch. These servers I have access to have all internet access blocked other then the SSH connection that I make in to the machines. 

Is there any bit of software that for example, can receive the piece of a block to solve on my main machine at home, and send it off to these servers through SSH to solve, then send it back?
What you describe is tunneling through SSH. Assuming you can connect to an SSH from the servers (outgoing TCP port 22 is allowed). A is the machine you want to run the miner on, B is the SSH server which we use to tunnel the traffic through (this has to be able to open connections to the mining pool) and C is the hostname of the mining pool you want to mine on, then it's a simple matter of doing the following on A:

ssh -L 8888:C:1234 B


and then pointing your mining software to localhost:8888. What this does is open a connection to B which may open connections to the mining pool and each connection to the local port 8888 will be transparently forwarded to C. Notice that you'll also have to change the port from 1234 to whatever your mining pool uses.