I confess to not having delved deeply into the history of the 14th Amendment's debates regarding the citizenship clause, and probably like most constitutional scholars, I simply assumed from reading the text of the provision that Congress did not have the power to deny citizenship to those born here to persons who did not enter with Congress' "permission" -- ie, legally.
But in starting to read up on this issue, I discovered that one of our leading scholars of immigration, Professor Peter Schuck, has argued for many years, starting in his 1985 book, Without Consent: Illegal Aliens in the American Polity (written with Rogers Smith), that Congress does have the power under the 14th Amendment "to regulate access to birthright citizenship for groups to whose presence or membership it did not consent." This past summer, Schuck and Smith published a long essay in which they summarized their views, which can be found here.
Of course, the question of whether Congress can legislate on this issue is a completely different question than whether the President can act unilaterally by executive order. Congress has legislated extensively in this area, and the President does not have the power to contravene Congress in this area, given the Constitution's explicit commitment to Congress of the powers over naturalization and other relevant powers Congress has that bear on this issue.
But given the discussions now emerging over the general issue of birthright citizenship and the original understanding of the 14th Amendment's citizenship clause, I thought readers would want to know of Peter Schuck's extensive discussion of this issue. I'm not endorsing Peter's views, of course, but I think many readers will want to be aware of them.
Update: After I posted this, Schuck and Smith published a Washington Post op-ed reiterating their analysis in short form, here.