ADStrider is right, Beatles' version of "Happy Birthday" does exist but I think most of the people just mistagged their song "Birthday" cause "Happy Birthday" is available only (as far as I know) on YouTube and, honestly, not so many people know how to download songs from YT. :p
A rare recording of Happy Birthday done by the Beatles. The recording is from an appearance on the BBC's 'Saturday Club'...which is who/what they are singing Happy Birthday to. At least that's what I've heard.