This has some great background:
Birthright Citizenship and the Constitution
One conspicuous departure from the language of the Civil rights Act was the elimination of the phrase "Indians not taxed." Senator Jacob Howard of Ohio, the author of the citizenship Clause, defended the new language against the charge that it would make Indians citizens of the United States. Howard assured skeptics that "Indians born within the limits of the United States, and who maintain their tribal relations, are not, in the sense of this amendment, born subject to the jurisdiction of the United States." Senator Lyman Trumbull, Chairman of the Senate Judiciary Committee, supported Howard, contending that "subject to the jurisdiction thereof" meant "not owing allegiance to anybody else . . . subject to the complete jurisdiction of the United States." Indians, he concluded, were not "subject to the jurisdiction" of the United States because they owed allegiance-even if only partial allegiance-to their tribes. Thus, two requirements were set for United States citizenship: born or naturalized in the United States and subject to its jurisdiction.
By itself, birth within the territorial limits of the United States, as the case of the Indians indicated, did not make one automatically "subject to the jurisdiction" of the United States. And "jurisdiction" did not mean simply subject to the laws of the United States or subject to the jurisdiction of its courts. Rather, "jurisdiction" meant exclusive "allegiance" to the United States. Not all who were subject to the laws owed allegiance to the United States. As Senator Howard remarked, the requirement of "jurisdiction," understood in the sense of "allegiance," "will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States."
That's in interesting article but it's argument is fatally flawed by the fact that once the legislation was passed the interpretation of the law was passed to the courts. The case used for illustrating this point is Wong Kim Ark. To oppose that you'd need a court interpretation (upheld) that an Indian was ruled not a citizen even though he was born not under the jurisdiction of his tribe (
"distinct political communities") and inside the US. Do you have one?
The individual in ELK V. WILKINS doesn't meet that standard because, "he does not allege that the United States accepted his surrender, or that he has ever been naturalized, or taxed, or in any way recognized or treated as a citizen by the state or by the United States. Nor is it contended by his counsel that there is any statute or treaty that makes him a citizen." AND
"Under the Constitution of the United States as originally established, "Indians not taxed" were excluded from the persons according to whose numbers representatives and direct taxes were apportioned among the several states," http://supreme.justia.com/us/112/94/case.html
As the court mentioned in that case, Indian tribes are dealt with by treaty/legislation and their members have their own jurisdiction outside the taxing districts of the states. This has nothing to do with illegal aliens.