Hi @Cras
I'd have to say extremely unlikely. This has nothing to do with Box Drive and more to do with the fact that Bridge does not work INTENTIONALLY with any cloud service. Some are able to use one cloud service and not others, while some cannot use any cloud service.
Bridge was made and designed to be used on images that are directly on a hard drive or a connected drive, not a cloud service. While this may change in the future, it is unlikely to change any time soon. But as always, YMMV. If you are able to make it work, great, but that is not intentional by the intention of Adobe.