UPDATE:
Henry used the box just a bit after I originally posted, and has used it often since. He never did go anywhere else, I guess he was just a bit nervous about the new surroundings.
My family just adopted a two-year-old Bombay from the local shelter, and he's been at our house for a day, and has yet to relieve himself at all. He's been eating a fair amount, and had some water and all, and seems relatively happy (he's in a "safe room" from the kids; with bed, litterbox...