Since the Confederacy existed for about four years, presumably some number of babies were born there during that time. Technically, those people were not born in the United States.
When those people reached the age of 35 (which would of course be after the war was over), were they eligible to run for president of the US?
Technically, they were born in the United States, at least under US law, and the law of any existing nation. The United States never recognised the Confederacy, nor did any other country. There are occasional wingnut groups who declare themselves separate from the United States, but a simple declaration of secession does not create a country – you also have to win the war.
The Supreme Court affirmed that the secessionist states were always an integral part of the Union in Texas v. White, 74 U.S. (7 Wall.) 700 (1869). Since they only had jurisdiction if Texas was a state at the relevant time (some time on or before January 15, 1865), by accepting the case, they implicitly affirmed that Texas was part of the Union. Further, in the judgement itself, they explicitly stated that the acts of secession of not just Texas, but every rebel state were “absolutely null”.
The requirement is actually not to be born in the US, but to be a natural born citizen. You can be born outside the US and still be a natural born citizen and thus eligible to be elected President. The relevant question is whether their parents (engaged in the secession) were still citizens of the United States and were thus able to pass on the citizenship to their children. I believe the answer from Dale M answers that question; no secession ever took place.
They could in theory be blocked under the 14th amendment in case they have '...engaged in insurrection or rebellion against the same [the United States], or given aid or comfort to the enemies thereof.'Given the young age of anybody born during the war, I deem that to be unlikely. It would however apply to adults engaged in the secession activities.
