I think you’re missing the point of the term “rightfully”. The people do have the right to know everything about a candidate that might affect his judgement or ability to do the job that he is elected to do. This would include any questions related to ethics, past associations, natural-born citizen requirements, etc.
Most employers conduct drug screening and background checks for this very reason. Many accountants have their credit reports reviewed before hiring. Small business loans require enormous amounts of personal info in order to determine if the recipient is worthy. An employer has the option (an should be legally bound) to require an employee to provide proof of citizenship or legal authority to work in the U.S. if it is in question.
Why should the most important job in the country have any different standards? As Obama’s employers, we the people have the right to ask these questions and require proof.