Our Constitution states that no person except a natural born citizen or a citizen of the United States--at the time of the adoption of this constitution--shall be eligible to the office of the president. Now, that seems to mean anyone not a citizen under the laws prevailing at the time of the constitution is ineligible to be president. The laws back then defined "citizen" rather narrowly, excluding many - including candidates in the 2016 election.
Moderator's Note: The intent is to require natural-born citizenship for eligibility. The supplemental eligibility served to grandfather-in founding fathers who were immigrants; the relevance of language expired with them - that's what the "or" is for.