I was looking through the app store on my iPad this morning and I was prompted to update Nook Kids. It looks like B&N has decided to go along with Apple's demands.
I can confirm that the link to the B&N ebookstore was removed. I checked before updating and there was a link in the upper right corner of the screen. It's not there anymore, and so far as I can tell that was the only change. Curiously enough, this wasn't an announced change. According to iTunes, the only differences were "minor fixes". But B&N also updated the listing with new screenshots; the ebookstore link is gone there too.
The Nook Kids app also shares one interesting detail with the Kobo app. You can no longer create an account from inside the app. I'm not sure it was there before,but right now I'm looking at the related Nook app (not Nook Kids) and I can create an account inside that app.
BTW, B&N haven't updated the Nook app yet, but they probably will push out the update some time this week. They've already changed 1 app, so they are probably already doing the same on the other.
On a related note, there are other companies responding to Apple's threats. The Financial Times chose to respond by launching an HTML5 app, and both Netflix and Hulu updated their iOS apps to remove any way to connect with their respective websites and create an account (much like the reading apps above).