Buchi is the cat that Haruka finds just after she escapes with Ayato from Tokyo Jupiter. Haruka tries to keep it as a pet but it escapes during their time in the naval base. However, Quon finds it but then gives it back to Ayato along with his luggage and from there on it stays as part of the family at Rikudoh's place. Haruka thinks it's the cutest thing she's ever seen.
This is the blue bird that Kunugi keeps in a cage; he named the bird after his late daughter. Though there isn't much to be said about this bird, it's interesting to note the symbolism it holds in the story. Time and time again, we see this blue bird flying around and often taking affection to Quon. We can safely assume that the director wanted to portray the story of the Blue Bird of Happiness in RahXephon. We see evidence of this when Ayato refers to the fairy tale when he was conversing with Kunugi by inquiring the names of the two children who were after the blue bird --Tyltyl and Mytyl. The message of the blue bird is that true happiness is usually found close to home, and that what is most important is the journey itself, not the destination.