Does anyone have a guess as to how this works? Perhaps the emails are viewed through a web browser? In any case, it would seem that the analog hole defeats the purpose of the security measure.
I think the way that they prevent it is that you'll get fired if you do it. People who work in the White House are usually people who a) want the sitting president to do well and b) want a career path that will get them to even better things than they're at now. A big Obama supporter isn't going to forward an email that might be embarrassing unless it's something like an order for human children to be served at a banquet. Likewise, if you want to someday become a congressperson or become a big(er) deal in DC, you don't want to get a reputation as "that guy that spews everything you don't want public".
So, really, you only need slight precautions to make sure that email isn't forwarded. The kind of precautions that would let a hacker easily bypass it, but you wouldn't accidentally forward something.
http://office.microsoft.com/en-us/help/HA101029181033.aspx
I have no inside knowledge, though, but I doubt they built their own system.
And you can still take a screenshot with a third-party program, and myriad of other ways to get around it. If you read it on the screen, you can forward it -- just takes a little more work.
The issue however, is what if someone unknowingly or accidentally forwards something the president wrote? There's easy-to-setup and use technology in pretty much all mail servers to check for and prevent that.
(I'd find it super neat however, if there were a program that took all of the president's emails and inserted extra random spaces/punctuation/etc. so that if there WERE leaks, they'd be identifiable.)
That would probably work well enough.