Every proper, normal, rational variety over an algebraically closed field is simply connected. Dmitri explains this via the smooth case and resolution of singularities, but this is true in any characteristic: see SGA 1, XI, Cor. 1.2.
EDIT: As Dmitri and Vesselin observe, the "proof" in SGA1 is sketchy, to say the least. One can argue as follows: if $X$ is our variety, there is a birational morphism $U\to X$ where $U=\mathbb{P}^n\smallsetminus Z$ and $Z$ has codimension $\geq2$ in $\mathbb{P}^n$. By the purity theorem, $U$ is simply connected. So any étale covering of $X$ is generically trivial (because its pullback on $U$ is trivial), hence trivial since $X$ is normal.
In fact, this proves that if $X$ and $Y$ (both proper and normal) are birationally equivalent, and $Y$ is regular and simply connected, then $X$ is simply connected. But this does not answer Vesselin's final question.
