I just re-read the original post, and I think I may have misunderstood something.
Are you unable to log into FaceBook, or just a specific game app in FaceBook, or using facebook to log into an iOS game? If the second, then your best bet is to go through FaceBook's support, or the developer's (if they have support). If the thrid, I'd ask the developer, gain.
Pretty much everthing recomended so far has been about logging into FaceBook itself. I'm not familiar enough with apps in FaceBook to make suggestions.
From what I can find out about Shuffle Cat (iOS version), it is currently a UK game only. When I follow a link (from Google Search) to the App Store, it tells me that it's only available in the Estonian app store.