I can answer the "why not" question. No hospital would make that claim nowadays because of HIPAA laws. It would leave them open to a whopping big fine and jail time.
Utter Rubbish.
Firstly, It is claimed by his cheerleaders that 0bama has already made his birth location public information.
Second, the hospital you were born in doesn't constitute "private health information". HIPPA provides confidentiality about your health condition. Presumably, we can safely assume 0bama was born somewhere, if he wasn't, then we have a whole different set of problems..