Besides going the proxy way and filtering all data, what about taking this to a lower level, and restrict devices from the network, change read/write permissions so no data can be stored on said computer, etc? Just a thought.
As far as using a proxy like linosaurusroot said, you would have to make sure that all data passes through a predetermined proxy that you set up, and yet, to see ALL files going through, you will have to sniff almost all of the data, etc. I'm saying this to try to see what else is a possibility in this case, as looking at all data 24/7 will take processing power, storage, etc, and possibly a better way exists.
Edit - If most of the files you suspect are image files, you could always run up driftnet - http://www.ex-parrot.com/~chris/driftnet/