Can you play Bingo Vlitz, Bejeweled, and Castleville on Kindle Fire HD?
As Justin said, if you can't find the app in the Amazon App Store or the Play Store, try another app market like 1Mobile Market to see if they have the Facebook connected games. If there aren't any apps at all, such as for Castleville, then you are pretty much out of luck. Trying to play Castleville through the browser (even if you have Adobe Flash Player installed and use a flash capable browser), most likely it will not work. These games are not optimized to be played through a mobile browser, and I don't think that they will be for quite some time. What makes it more difficult to play these games through any mobile browser is the touch interface; I have tried playing Facebook games in mobile browsers on my Android phones, and if you have to "click abd drag" anything, the mobile browser moves instead or it isn't recognized in the game.
BINGO Blitz for Android/Kindle Fire
Bejeweled 2 to bedazzle Kindle Fire along with Scrabble, Monopoly et al
you can download the apk to desktop then trnasfer to sd card then plug it to kindle then try to run from there
If you can't find an app for the game in the Amazon App Store, the answer is probably no.