Sorry I don't know about the boppies (never noticed one but I never asked either).
As far as the baby care centers go, I loved the one at Animal Kingdom. It had private nursing rooms where you could shut the door and they had changing tables and rocking chairs in them in addition to the changing room (a room with 10 or changing tables in it). They also have a bathroom and I think another "play room" where a little one could run around a bit.
The Magic Kingdom one was older and dimmer. The nursing room was shared (all nursing moms in one room with 2 doors). It was also much busier when I went there than the other 2 were. This was my least favorite.
MGM had private nursing rooms again, but I think it only had a curtain to pull shut.
I did not breastfeed her at Epcot, so I don't remember what the nursing room was like there. (I BF but also use formula when we are out and about--I didn't starve the baby on the Epcot day

).
I loved having these centers, but selfishly wish there was more than one in each park. It was a nice, quiet, cool place to feed & change the baby, but not always very close when I needed it.
I did manage to BF during a few dark shows (the 3Dmovies, Laugh Floor Comedy Club for example). I sat in the back row (by the handicapped areas) and treid to be discreet as possible).
We also were happy to find pacifiers, diapers, medicines such as tylenol, onesies and even bibs there, all for a price of course--but if you need any of these items while you are out at the park believe me, you are willing to pay
any price.
Have fun!