I love the sound of this project, but...
What kind of footage do you want to capture: wide-angle, full court or player-focused zoom? Will you be using just the one camera or many and will they be stationary or moving? Do you need to overlay any visual effects onto the stream? Will you have a stable broadband internet connection?
These and a dozen more questions could impact the kind of equipment and workflow you're going to need to pull off a live stream like you described.
At the very least, you'll need a camera or camcorder capable of providing a clean-feed output via composite, HDMI or SDI and a means to capture that feed to a PC/Mac for encoding.
In terms of cameras, I'm not convinced the Canon 600D is your best option; I'd choose the Panasonic GH2 over that almost any day, but especially in this situation.
As for capture and encoding, I'd recommend taking a look at the options available from TeraDek (excellent, wireless solution for capturing camera footage and streaming it to the internet) and Blackmagic Designs.
Good luck and keep us updated with your progress!