The very first Republican Party nominee for President (in the election of 1856), John Fremont, was born in the U.S. but he was the child of two aliens who never became citizens. No one at the time questioned his eligibility.
Are you certain? I'm pretty sure his mother was American.
His father, though, was clearly a Frenchman who was not a naturalized US citizen, and had no intention of ever becoming one. He was planning to take his family to France when he died.