Well, it looks like I sold myself short. I was able to create a poor man's hack of a Box client. Improvements welcomed, but in the meantime, it works.
From the brief readme I created:
Purpose: Search your Box.com sync’d folders for files, and when selected copy the link of that file to the clipboard for easy sharing.
For the app to work, the files must be stored on your machine and you must be running the Mac Box client.
Usage: On first use, click the “File Filter” box and set the search scope to only search your Box.com sync’d folder.
The way the app works is to simulate mouse clicks that bring up the context menu in Finder and manually select the Box menu and the items underneath that menu that copy the sharing links to the clipboard.
If it isn’t working for you, review your own context menu when you right click inside Finder and make sure that the Box menu is appearing.
You can download it here.