That baby was born with a sunTAN.
He said he wants to prolong his birthday so he gets presents forever.
If you want to say Happy Birthday, you'd just say 良い誕生日！ （よいたんじょうび！）.
However, if you're asking someone what day is their birthday, you'd add 'の日’ like this:
the birth of (figurative) (i.e. Miles Davis' "birth of the cool" album)