If you just want to update the coordinates for the rectangle that was drawn using mouse events, you could do something like this: http://jsfiddle.net/HA8aP/2/
It adds a name parameter to Box2, and the name for the mouse drawn box will be "initialBox". Then the name is validated when dragging. In this example, all of the double-click created boxes are called "otherBox", but you could just as easily give them all unique names in the addRect function and then generate unique coordinate displays for each one.
if (mySel.name == 'initialBox') {
$mouse.innerHTML = "(" + mySel.x + "," + mySel.y + ")";
document.getElementById("TextROI_ULx").value = mySel.x;
document.getElementById("TextROI_ULy").value = mySel.y;
}