Chip,
You're in "Opinion Country," probably why the rules aren't carved in stone. Flickr has
200,000+ photos in a group here:
http://www.flickr.com/groups/squareformat/
If you look through enough of these, you'll begin to see it isn't much different from non-square. Circular subjects (notably blooms, containers, animal faces) seem obvious choices, but this group just puts anything that they believe looks good in a square.
I have about 75% square crops on Flickr, mostly Macros that just seem to fit squarely with my subjects. You can also rotate a square and stack it with other squares without any worry about aspect or orientation.
I'd likely crop your faucet in portrait, and might have taken the shot so the drop and spout were about 30 degrees turned to the left, leaving some oval shape to the handle. And that's how it goes. We make our choices based on opinion most of the time. I like square, so my answer is more positive. (I might also square crop your faucet much closer so the round handle is more "circle-in-a-box."
You can still use the rule of thirds and most other compositional suggestions, including diagonals being more interesting. The faucet seems to enjoy the verticality of reality; flower stems like a good diagonal. More opinion.