What would be the best approach to creating an application on Heroku that has the capability to push data to the client?
I have looked at Juggernaut; however, it requires Flash and I am not sure about it actually running on Heroku.
Any help would be appreciated.
Yes juggernaut is not supported by heroku. I guess you should get in touch with the heroku guys for this. Unless and until heroku doesn't make it an addon package you won't be able to use juggernaut.
although I'm not using Heroku personally, they've just announced experimental node js support few days ago. While they are working on it, you might want to explore Node.js and Websockets in the meantime:
http://devthought.com/blog/2009/12/nodejs-and-the-websocket-protocol/