How can I get hold of the an extender object in the client?

If I'm understanding correctly that you're hoping to get a reference to the behavior from script, check out how the TextBoxWatermark and PopupControl implement an ID property which causes Atlas to be able to access them via Sys.Application.findObject(...).
If you want to catch a button click on the server side, why can't you use a regular asp:Button control?

Thanks David,

That's exactly what I needed, I got it working now.

Thanks for the replysburke_msft,

I forgot to say in my post that the control works in the client side, but thanks anyway.