The Fourteenth Amendment does not use the phrase natural-born citizen. It does provide that "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."
In Rogers v. Bellei the Supreme Court only ruled that "children born abroad of Americans are not citizens within the citizenship clause of the 14th Amendment."
Good citation, if off point somewhat. The SC held that the Congress has powers to decide what makes a "citizen" when not born on US soil. The residency requirement is moot in this discussion, but makes for interesting reading--especially Black's dissent. Congress determined an affirmative action is required; dodging the draft was anti-affirmative, so citizenship was refused by Bellei due to that action. The reasoning here was dual citizenship which is unlawful for US citizens. And Bellei had only one US citizen parent.
As to your comment about McCain: he was born on a United States Military base of two native born Americans, one of whom was serving in his country's Armed Forces. Hard to get more 'US' than that. His later life, and heroism as a POW, more or less renders any discussion on his status moot, one could conclude.