If you go to the search option and type in Happy Birthday and then ask it to search only the translation & vocabulary section, you will see many different post stating how to say it. There is even one post that translates the Happy Birthday song.
"Šťastný Narozeniny" is the literal translation of "Happy Birthday"; however, no one says it that way in Czech. Use Ajik's translation.
Yeah, just say Vsechno nejlepsi or Vsechno nejlepsi k narozeninam (and a name of the person) Sorry for not using hacek and carka (the signs above the letters) I don't have Czech letters on this computer. Please, can you next time use search for translating such basic phrases? It's really a basic phrase and there's lot translation of it in the forum.