Are you thinking of a specific President or Presidential candidate? As far as I know, no President was born in the Confederacy for the simple fact that following the Civil War there were no Southern Presidents until Wilson, who was a "transplanted Southerner" that lived primarily in New Jersey. The next Southern President, Truman, was too young to have been born during the Civil War, and he, anyway, came from Missouri, which remained loyal to the Union.

There weren't even any Presidential candidates from the South. This is partly because the Republican Party was a politically non-viable organization in the solidly Democratic South. But the Democrats didn't nominate Southern men either, preferring New York men like Tilden or Cleveland, or Northern war heroes like Hancock. The bloody shirt of war still pursued them after all, and nominating a Southerner would probably open them to attack as being rebels when the Party was trying to move away from its old association with slavery. This did not stop them from electing former Confederates in the South, but national politics wouldn't allow to nominate a Confederate. Wilson, again, comes closer, having Confederate parents and having lived some of his childhood in wartime Georgia, but he was born in Virginia previous to the war and lived in the North.

In any case, the issue of citizenship would be moot because the position of the Union government during the war and Reconstruction was that secession was void and null, the states never left the Union, and that this was a rebellion of individuals. Thus, for legal purposes, all Confederate territory was considered to be United States soil under occupation by insurgents. The 14th Amendment granted citizenship to all people born in the United States, and since the position was that the Confederacy never legally existed it too was American territory during the war. So anybody born in the Confederacy would be an American citizen. To consider the Confederacy a foreign country in any way or form would go against Union policy and war objectives.

Citizenship, in any case, was not stripped except from high ranking Confederates like Lee or Davis. Whether it was ever stripped from the common soldier it's not clear since, again, the official position was that the states had never left the Union and thus they had always remained US citizens, subject to the authority of the government. To say that they were foreigners would be to recognize implicitly that the Confederacy legally existed as a foreign nation. The terms under which the Confederate armies surrendered also implied that they were still citizens. Grant certainly understood the terms offered to Lee along those lines, arguing that treason trials would be a violation of those terms. If there were any doubts, Johnson's amnesty proclamation restored the citizenship of almost all Confederates except some high ranking officials, who then could apply for individual pardons that were almost always accepted.

In any case, this wouldn't apply to people born in the Confederacy. Citizenship was not revoked per se, but suspended. The confusion may arise from different conceptions of citizenship. 19th century Americans understood citizenship as the privilege to be part of the body politic and, if certain requisites are fulfilled, of exercising the privileges of voting and being elected. So you could be a citizen and still have no right to vote. Think of how children nowadays can be citizens but they can't exercise the full privileges of citizenship yet, or how women back then were not considered citizens because they weren't part of the public sphere of men. The citizenship of men like Davis and Lee was revoked in the sense that they could not exercise the privileges of citizenship, but this does not mean that they were seen as foreigners or stateless, or be deported like it happens with non-nationals nowadays. They weren't citizens anymore, but they were still Americans.

This punishment was a result of their actions, and obviously a newborn or a toddler can't be held accountable as traitors. So both legally and in the perception of most Americans, people born in the Confederacy were as American as anybody else, and they would enjoy full rights of citizenship because they did nothing wrong, being, you know, babies when the rebellion was still underway. So, yes, a person born in the Confederacy could have technically become President but it was very unlikely due to the politics of the period, not because of laws.