Sven:
THe story, as I have heard it over the years...
"Handi" box was christened as it was usable for any single device, or as a "pull point", and it was cheaper than the 4" square.
I have seen them used for all types of things, mostly not legal per the NEC. Multiple cable / conduit entries, and a device stuffed in. They still show up on homeowner installs; most EC's don't use them.
Also, you don't need a "plaster ring" or raised cover to mount your device with a "handi box", which results in a "cheaper" install.
Yes, there is a masonry box (mud box) in many configirations. The easy way to tell is they have square corners.
John
