The 14th Amendment does not guarantee citizenship to children of illegal aliens who were born in the U.S.: that's only the way that it's been interpreted for several decades.
Here's Section 1 of the 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. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
Now, here's an 1866 quote from Senator Jacob Howard concerning that Section:
"Every person born within the limits of the United States, and subject to their jurisdiction, is by virtue of natural law and national law a citizen of the United States. This 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, but will include every other class of persons. It settles the great question of citizenship and removes all doubt as to what persons are or are not citizens of the United States. This has long been a great desideratum in the jurisprudence and legislation of this country."
And, here's 1997 Congressional testimony from Edward J. Erler (California State University, San Bernardino; The Claremont Institute for the Study of Statesmanship and Political Philosophy):
Clearly, the author of the citizenship clause intended to count "foreigners," "aliens," and those born to "ambassadors or foreign ministers" as outside the "jurisdiction of the United States." Senator Howard knew, as his reference to natural law indicates, that the republican basis for citizenship is consent. This is the natural law principle of the Declaration of Independence that proclaims that legitimate governments derive "their just powers from the consent of the governed."
Rep. Tom Tancredo (R-Colo.) has a bold idea to stop illegal immigration: Deny automatic citizenship to the children of undocumented immigrants. "There is general agreement about the fact that citizenship in this country should not be bestowed on people who are children of folks who come into this country illegally," he told the Washington Times. General agreement? Perhaps among Mr. Tancredo's friends in the House but not among the framers of the 14th Amendment. Indeed, any such modern consensus would have a small problem in the text of the Constitution, which is, inconveniently for anti-immigrant demagogues, not subtle on the point. The 14th Amendment begins: "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." Not "all persons except children of illegal immigrants," not "all persons except those Congress exempts in moments of nativism." All persons.