A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative logarithmic 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 logarithmic Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale 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 logarithmic Kodaira dimension, admits families of non-proper étale 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.