... is primarily about citizenship and whether or not a state can restrict rights of a citizen, and whether or not Missouri did.
Virginia Minor was a natural-born citizen, who was also a woman. She wanted to vote in Missouri, which had a law that only men could vote (a great idea, btw! ;-)
The registrar (Happersett) denied her registration and she sued.
The US Supreme Court identified her as a natural-born citizen because she was born in the US to US citizen parents. They said there were other forms of "citizen" but they did not need to address those other forms since Minor was born in the US to US citizen parents.
They went on to say that voting was not a right, but a privilege and a state could determine who could and who could not vote. It was OK for Missouri to deny women the vote because voting was not a right held by all citizens.
Of course since then, women have the right by way of amendment to the Constitution. This case was decided in 1872, before the 19th Amendment.
But the case ABSOLUTELY DOES say that it has long been understood that a natural-born citizen is one who is born on the soil to citizen parents:
"At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners. Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents. As to this class there have been doubts, but never as to the first. For the purposes of this case it is not necessary to solve these doubts. It is sufficient for everything we have now to consider that all children born of citizen parents within the jurisdiction are themselves citizens."
This is the US Supreme Court validating that a "natural-born citizen" is a person born on US soil to 2 US parents.
The problem for the deniers is that they have ZERO evidence to back up any other claim.