I don't think I've ever seen a Vattel-Birther even so much as acknowledge this quote from Hamilton. Replies that are made to a post containing it just skip past it. The "I'll just pretend it doesn't exist" technique. DiogenesLamp is an exemplar of that technique.
It undermines their arguments on several levels.
The Brits didn't have "Citizens", they had "Subjects." As the condition of a subject is completely incompatible with the condition of a citizen, putting forth a claim of ownership on someone simply because they were born on the King's soil is not compatible with the principles of American independence.