Just in case it occurs to anybody to try this here:
Server push means a continuous socket connection. That means that the browser needs to support WebSockets, and that the server needs a WebSocket server (a daemon, which can be a continuously-running PHP script).
The requirement for WebSockets pretty much brings the browser choices down to current versions of Firefox and Chrome. (Opera and Safari support an older, insecure version of the protocol; IE doesn't support WebSockets to date.) The requirement to run a daemon -- a script that runs continuously -- means that you won't be able to run the server side of the socket on either a Free Hosting account or a Premium account (although you should be good to go on a VPS if you don't mind burning your fair share cycles to run the daemon). There's no way to do push without a daemon -- the server needs to monitor the chat data, and it needs an open connection to the browser to send the data. Each browser that's using chat needs its own socket, and that means clock cycles, memory and connection-maintaining chatter over the TCP connection. It is far more intensive to implement than client-pull. (And that's why Skype is partially peer-to-peer routing -- it takes a lot of load off of the server.)