I was looking into this a little bit and it seems that there is no special event that fires when the X is clicked. However, the click event for the input field is triggered. So, the workaround appears to be checking if the field is empty, but the view model value is not. Then, setting the view model value to be empty.
Here is a sample with a possible implementation of a "reset" binding to make this happen:
http://jsfiddle.net/rniemeyer/vrbzv/ (will only show the "X" in Chrome). This doesn't fix the reset button, but does fix the "X" on the search field.
Hope this helps.