We'll I'll consider the question posed in Glasby's answer. Take $n=3$ and $\pi(x,y,z)=( (x-1)^2 - y^2, xy, xz )$. To see that $(u,v,w)$ lies in the image, note that there is always at least one pair $(x_0,y_0)$ such that $((x_0-1)^2-y_0^2,x_0 y_0) = (u,v)$ and $x_0 \neq 0$. Then $\pi(x_0,y_0,w/x_0) = (u,v,w)$.

Thus there exist surjective quadratic maps for all $n \neq 1$. This doesn't really answer the original request for a criterion though.

I'll consider the question posed in Glasby's answer. Take $n=3$ and $\pi(x,y,z)=( (x-1)^2 - y^2, 2(x-1)y, xz )$. To see that $(u,v,w)$ lies in the image, note that there is always at least one pair $(x_0,y_0)$ such that $((x_0-1)^2-y_0^2,2(x_0-1)y_0) = (u,v)$ and $x_0 \neq 0$. To see this note that $(0,y_0) \mapsto (1-y_0^2,-2y_0)$ and $(2,-y_0) \mapsto (1-y_0^2,-2y_0)$. Thus $\pi(x_0,y_0,w/x_0) = (u,v,w)$.

Thus there exist surjective quadratic maps for all $n \neq 1$. This doesn't really answer the original request for a criterion though.