this will happen
1. EO nullifying Constitution regarding ius soli
2. goes to supreme court
3. kavanaugh and gorsuch oblige
4. EO are now superior to constitution
5. EO nullifying... freedom of press?
Steps 1 through 3 are what would happen, but step 4 is not why. There's a path to rule in Trump's favor that doesn't involve trying to argue that an EO can directly overrule the Constitution.
A ruling on the 14th Amendment if Trump tried to do this ridiculous thing would be centered around specifically this part of the sentence granting birthright citizenship:
"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."
What that clause is there for is to specifically exclude the children of foreign diplomats, who are for the most part outside of the jurisdiction of the United States (hence diplomatic immunity), or the children of a hypothetical foreign army that has invaded American soil. That way if, say, the Mexicans invaded Texas and sent up military bases on US soil, the children of those soldiers would not automatically become US citizens.
The white supremacist argument is that because illegal immigrants are an invading force and are not authorized to be here they qualify as being not under the jurisdiction of the United States. That's a nonsensical interpretation of the text, of course, but it's what they're going with. All they have to do is make the Court agree with their bizarre redefinition of jurisdiction (which would mean breaking with precedent), and the executive order would be perfectly legal and in court-approved compliance with the Constitution.
When Trump says "they" are telling him he can change this with an executive order, this is precisely the scenario that "they" are envisioning.
