Well, we have a clue to their origin.
Jim Carlton astutely inquired about whether there were any markings on the cap.
Eric searched more thoroughly today, and found a faded word on the top of the black cap:
When Eric searched for the word "KURE" on the Internet, he discovered that Kure is a port town in Japan (near Hiroshima).
And then he started searching for KURE products. Here's what he found:
Lots of black plastic caps! KURE is a company that makes cleaning products and lubricants — see their website here. (Note that their slogan is, "Let's Move!" It appears that the mystery anemones took their slogan literally!).
The fact that the anemones were growing in a KURE product cap makes us think that they might have originated in Japan rather than the Indo-Pacific. We'll continue to follow up on the identity of the mystery anemones.