How can I pass arguments to anonymous functions in JavaScript?

Your specific case can simply be corrected to be working:

<script type="text/javascript">
  var myButton = document.getElementById("myButton");
  var myMessage = "it's working";
  myButton.onclick = function() { alert(myMessage); };
</script>

This example will work because the anonymous function created and assigned as a handler to element will have access to variables defined in the context where it was created.

For the record, a handler (that you assign through setting onxxx property) expects single argument to take that is event object being passed by the DOM, and you cannot force passing other argument in there