Aha, that seems to do the trick:
So for the record, the only other thing I did was run this on BoxA:
ssh-keygen -e -f new-rsync-key
...to export my public key in the right format. I copied the output of that to the keys tab of UserMe in the users section on BoxB in the webui, where it sits alongside UserMe's proper password-protected SSH key.
So far so good! Thanks subzero79!