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.