Who counts as a citizen of the country is a framework question. Does the 14th Amendment confer birthright citizenship on the sons and daughters of all immigrants, even those illegally present? Or does the ambiguity of the concept of "jurisdiction" leave room for laws defining natural citizenship less broadly? The puzzle's solution depends on what method of constitutional interpretation you apply.
Our issue is interpreting the first clause of Section 1 of the 14th Amendment:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
The ambiguity of "jurisdiction"—a context-bound concept in U.S. law—sets the problem. I earlier concluded that textualist interpretation offers much to recommend jurisprudentially, as it comports with American Constitutionalism. Using essentially textualist methods, the U. S. Supreme Court held for the broadest birthright-citizenship in dicta in the thorough opinion United States v. Wong Kim Ark (1898) 169 U.S. 649. The Wong Kim Ark court found the term's meaning authoritatively interpreted in the earlier decision The Exchange (1812) 7 Cranch. 116. (See Wong Kim Ark, supra at p. 683.) Note the rule-governed, text-based nature of the interpretation in this passage:
By the civil rights act of 1866, 'all persons born in the United States, and not subject to any foreign power, excluding Indians not taxed,' were declared to be citizens of the United States. In the light of the law as previously established, and of the history of the times, it can hardly be doubted that the words of that act, 'not subject to any foreign power,' were not intended to exclude any children born in this country from the citizenship which would theretofore have been their birthright; or, for instance, for the first time in our history, to deny the right of citizenship to native-born children or foreign white parents not in the diplomatic service of their own country, nor in hostile occupation of part of our territory. But any possible doubt in this regard was removed when the negative words of the civil rights act, 'not subject to any foreign power,' gave way, in the fourteenth amendment of the constitution, to the affirmative words, 'subject to the jurisdiction of the United States.'
(Wong Kim Ark, supra at p. 688.)
Notice the use of analytic rules, a kind of common law of interpretation. In construing "jurisdiction" the court solves the puzzle like an equation, substituting a definition obtained from an earlier case. This is excellent textual analysis, but as authority, it is dicta: Wong Kim Ark found that one foreigners' son was not a citizen because he was born outside the U.S.
The Wong Kim Ark dictum wouldn't deter an Originalist; he observes that the 14th Amendment, designed to grant citizenship to slaves, did not have the offspring of illegal immigrants in contemplation: the United States hadn't yet passed any immigration laws. Strong Originalists are loathe to restrict the power of Congress when the legislation forbidden was outside the originators' contemplation, but omission of any protection not in the original contemplation is an extreme Originalist variant, which struggles hopelessly against vagueness, as it supplies no guidance on how much information about a future event must fall outside the originators' anticipation to disqualify. Professor Richard A. Posner proposed a more moderate Originalism, according to which the judge must place himself in the position of the originators: what would the originators advocate if they could foresee the new conditions, assuming the least historical modification yet placing the matter in the judge's purview. Using this approach, a court could imagine what the originators would have provided if immigration restrictions had existed at the time of the 14th Amendment's promulgation. An Originalist judge would likely rule that the patriotic originators, having won a war to preserve the Union, would not have intended to provide a law-breaking incentive which yielded no offsetting gain for the extant inhabitants. Such a judge would be likely to rule that an illegal immigrant's child is not—in the sense the proponents and ratifiers intended—born within the "jurisdiction" of the United States.