Here is what the constitution says on the matter who is eligible:
"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 President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States."
Yes, I believe that that's what I said above. Though, again, I don't know what the historical precedent is for interpreting the "natural-born citizen" requirement with regards to people who were previously citizens of a sovereign country that later became a US state. The best example I can think of would be Vermont, which joined the Union after the Constitution was adopted but had never been part of the Union under the Articles of Confederation. I don't know if the question of whether or not citizens of the formerly independent Vermont, who not been U.S. citizens at the time of the adoption of the Constitution, were retroactively interpreted to have actually been US citizens by virtue of Vermont having been covered under the Declaration of Independence or not.
In the first part of the miniseries, they explained it was allowed due to a clause in the treaty that oversaw the merger.
Yeah, but the problem is that a treaty cannot trump the US Constitution. The Constitution is the supreme law of the land, and any piece of legislation, whether an Act of Congress or a ratified Treaty, which is found by the US judicial system to conflict with the Constitution, is nullified.
Of course, by the same token, the US Constitution also requires that there only be one United States Representative for every 30,000 people, but so far as I know, that particular clause has never been fulfilled; instead, they just try to make sure that every Rep's district is roughly equal in population.