The 14th Amendment issue is very interesting to look at. From the Wong vs US ruling (that actually addressed children of lawful permanent residents and not those here illegally), to the intent of the 14th- it'll be interesting on how this plays out. FTR, I think this one is logically solid in theory and for what the intent of the Amendment was meant to do, however, in practice birthright citizenship as we know it won't change. I think it's wasted effort.
The phrase most in question is "subject to the jurisdiction thereof"... if you're here illegally, you aren't under the jurisdiction as such, you're under the jurisdiction and laws of your home country. There are already exemptions in the ruling (enemy combatants, foreign dignitaries), and the EO, as written, clarifies and expands those categories.
If you're Canadian going on vacation to the Bahamas and have a layover at JFK, and during that layover you have a baby- that baby is American thanks to the 14th, right?
That's the argument from the folks arguing for what birthright citizenship has become, not the intent of those who wrote it. It's also super fun to see the left try and square the circle of saying things like "Borders and citizenship are just imaginary lines on a map and a construct" and then argue for those magical lines to count and citizenship to be applied.