  A smooth complex variety satisfies the Generalized Jacobian Conjecture if all
its \'etale endomorphisms are proper. We study the conjecture for
$\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that
$G$-equivariant counterexamples for infinite group $G$ exist if and only if
$G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat
polynomials. Taking universal covers we get rational simply connected
$\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper
$\mathbb{C}^*$-equivariant \'etale endomorphisms.
  We prove also that for every integers $r\geq 1, k\geq 2$ the
$\mathbb{Q}$-acyclic rational hyperplane $u(1+u^{r}v)=w^k$, which has
fundamental group $\mathbb{Z}_k$ and negative Kodaira dimension, admits
families of non-proper \'etale endomorphisms of arbitrarily high dimension and
degree, whose members remain different after dividing by the action of the
automorphism group by left and right composition.
