UPDATE: Wamo has even more details about the encounter and the pet over on the WarcraftPets forums!
Someone found and killed the Darkmoon Rabbit on the beta, and it did indeed drop the pet!
Looks like the rabbit is a neutral mob and it spawns in the Darkmoon Cave on the faire island. Not sure of its spawn times but I'll be sure to check regularly now that a spawn point has been confirmed.
With 72 million health, it looks like we're going to need the help of a party or even a raid. I wonder if the pet that it drops will drop for everyone in the group/raid. Or if there will only be one drop and everyone will have to roll against each other. Last I checked, the Darkmoon Rabbit companion is Bind on Use, so we'll probably see it on the AH sooner or later going for a large chunk of gold.
Strangely, though, this companion isn't listed in the Pet Journal. Perhaps rarespawns don't show up in the journal, so as to keep their obscurity intact and keep them "rare"? That's just my guess so far. The Pet Journal's incomplete; the Darkmoon Rabbit might be listed in there in a future beta patch.
Anyway, check out the Darkmoon Rabbit kill thread that I linked. The rabbit pet is... PINK! Or at least that's what it looks like to me. And only for now (its tag says "Placeholder Art" so we can assume it'll be updated and changed in a future build). It's not the bright, hot pink bunny that was datamined just before Noblegarden was first released. Instead it looks like a lighter shade of pink, but pink is pink. Can't complain? :P
Personally, I'm inwardly cringing at the thought of collecting this particular companion. One, since I'm guessing it will be like a rarespawn, it will be highly sought after. I can imagine many people camping the spawn point and fighting to tag it (on PVP servers). Two, with so much health, you'll definitely need some help killing it, so take those rarespawn campers and multiply by 5-10 or more because they'll likely enlist the help of friends/guildmates. Three, the Darkmoon Island is only accessible for one week at the start of every month. I'm not exactly a huge fan of the limited time idea, but at least the pet is guaranteed to drop (I think?).... IF you're lucky enough to tag the mob, kill it before you're zerged by other players, and finally win the roll on the pet.
A lot of "ifs" are involved... more than my liking. It's not my ideal idea of "fun" but we'll see. Maybe it won't be as bad as I'm imagining it. I'm sure there will be some who enjoy this type of encounter at least.
PS: If this is an April Fool's joke, I totally fell for it lol. It seems totally plausible though. :P That's my story and I'm sticking with it!