How to overload operator equality for JavaScript objects

You can't overload ==, but == has an implicit .toString() call, so whatever .toString() returns will allow you to effectively overload == (kinda):

function foo(){}
foo.prototype.toString = function(){ return 42; }

var x = new foo();
x == 42; // true

As for how to do this in Dojo, I don't use Dojo, sorry, but the gist is that you get a reference to whatever object is creates and add thatObject.prototype.toString as in my example.

You can't. JavaScript doesn't support operator overloading.

You can't in Javascript/ECMAscript. You can overload operators in ExtendScript from Adobe. See this example. Also see this blog entry (pro), or this (contra).