element.boxObject.x/y/width/height can mostly be replaced with getBoundingClientRect(). It is almost identicial, although its behaviour in scrolling areas is different. If performing computations with the values, current code might add on scroll offsets and the like, which probably won't be needed with getBoundingClientRect.
element.boxObject.screenX/screenY can be replaced with element.screenX/screenY.
The property related functions would probably work with just properties on the element. I checked at one point and Thunderbird doesn't use them as far as I can tell anyway.
The element relation properties (at the end of BoxObject.webidl), have no exact equivalent but again I'm not sure if Thunderbird needs them. If so, I can suggest alternatives.