No, their place of birth would not make any difference. They were citizens of the United States, not Citizens of the Cherokee Nation.
It's possible you may have an ancestor who was Scottish or German, but when they left that country to live elsewhere, they gave up their Scottish or German citizenship and became a Citizen of the country they moved to.
The counrty they left would not maintain any further records of them, but they would still retain Scottish or German ancestry.
That's how it works with Cherokee/American Indian People.
Unforunately, there has been a lot of inaccurate, erroneous and misleading information posted on the internet.
Especially, regarding American Indian People.
I forgot to include in my initial reply that the Henderson Roll was taken in 1835.