Using Hauppauge HD PVR and Mac Mini to Stream Game Video
Argh, I wish I had seen this in time for the bounty.

I do broadcasts using Camtwist. Camtwist will become a video source and it does things like just broadcast the desktop. So if you have EyeTV maximized and Camtwist broadcasting the desktop and your flash broadcasted using camtwist as a source, you should be set! If you want audio, soundflower is good for that, but not quite as straightforward as Camtwist. Might have to play with it. But there are lots of good tutorials on this online, give it a google.