The statement is false. I think I learned of this example from James Borger on a blog, but I'm not sure. If you take a nodal cubic (notably quasiprojective), there is a flat, unramified cover by an infinite connected chain of copies of P^1, each glued transversely to its successor at a point. This is not profinite. If I'm not mistaken, the etale fundamental group of the nodal cubic over a separably closed field is $\mathbb{Z}$, not its profinite completion.
S. Carnahan ♦
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