+1 to what confuse said...
In addition to being finicky eaters, they can be quite shy as well.
I purchased one in the past and threw him right in w/ all my other fish. He was too shy to come out. When it was feeding time, he'd try to sneak a bite in, but as soon as the other fish started to converge on the same bite, he'd dart right back into the rocks. He ended up dying in maybe two weeks.
About 6 months after that, I decided to give it a go w/ a new tactic. This time I bought two of them and threw them both in a new frag tank that had no other fish in it. I'm happy to say they are both alive and eating quite well. They ate right off the bat, but were still a little shy. Now, they're almost to the point where they aren't even afraid when I put my hand in the tank to muck w/ the frags.
One thing to note, I do think getting a pair is helpful because one of them didn't eat pellets at first (only mysis). The other one, however, went to town on the pellets. After seeing that one eating up all the pellets, the other one started to as well.
Definitely two of my favorite fish. Good luck!
EDIT: To clarify, I have borbonius anthias. I'm not sure what region they are from and/or if the ones I have are different than the Japanese ones you are referring to. They are pink w/ yellow spots.