The 14th amendment was not passed until 1868. Until then, there was not a single restriction on Congress's ability to define citizenship any way it wanted to. Further, the 14th amendment with regards to citizenship, only limited congress's power by reinstating jus soli.
So other than the 14th amendment which specifically addresses citizenship, and the other more general provisos of the various amendments (equality clause, etc), the naturalization clause of Article I Section 8 is without restriction.
”. . . I deny that the Congress of the United States ever had the power [emphasis added] or color of power to say that any man born within the jurisdiction of the United States, and not owing a foreign allegiance, is not and shall not be a citizen of the United States.” - John A. Bingham, (R-Ohio) US Congressman, Architect of Section 1 of the 14th Amendment, March 9, 1866. Cong. Globe, 39th, 1st Sess., 1291 (1866), Sec. 1992 of U.S. Revised Statutes (1866), Cf. U.S. Const. XIVth Amend. (See top half of 2nd column)