Their argument -- best summarized here by George Will -- is that there's no need to change the 14th because Congress has it in its power to remove birthright citizenship via legislative action. Basically, they believe that the seemingly plain language of the 14th doesn't actually mean what it has been understood to mean by courts and governments for the past 140 years. They base this revisionism on their
(re-)interpretation of the phrase "subject to the jurisdiction thereof", which has previously been held by the Supreme Court to exclude only children of diplomats and Native Americans.
And then, of course, there is the far less radical -- or far more pusillanimous, depending on how you view it -- Hearings Birthrighterism, which is the position of Boehner, McConnell, McCain, et al, supporting unspecified hearings that they hope will placate the Crazies without actually committing to, you know, doing anything.
As it happens, I don't think GH interpreted Keyes correctly. My take on Keyes remark was actually that they were dramatically crazier than any of the opinions on offer -- basically that God had restricted Congress's right to change birthright citizenship, either through legislation or constitutional amendment. But the upshot was that birthright citizenship cannot be touched.