Let $k$ be a field. In what generality is it true that higher etale homotopy groups 
of $\mathrm{Spec}\,k$ vanish? 


If the absolute Galois group is finite, we have a universal cover $\mathrm{Spec}\,k^{sep}\rightarrow\mathrm{Spec}\,k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $\pi\_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$. 


I am not sure what happens for general fields. 



Do higher etale homotopy groups of spectrum of a field always vanish?
The étale topos of a field $k$ is just the topos of sets with a continuous $\mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $\mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $\mathrm{Gal}(k)$.


Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(\mathrm{Spec}\,k)\_{ét}$ is the homotopy colimit of $B\mathrm{Gal}(k)/H$, and so it is the profinite space usually written $B\mathrm{Gal}(k)$ or $K(\mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.


