Both curves have no rational points.
Curves of this shape map to genus 2 curves of the form $y^2 = a x^5 + b$ (one can make $a = 1$ if one likes), by quotienting out by the group of automorphisms generated by $(x:y:z) \mapsto (\zeta x:\zeta^{-1} y:z)$ where $\zeta$ is a primitive fifth root of unity (one can pick any two coordinates for the scaling to act on). For curves of this type, Magma can fairly easily compute the Mordell-Weil group (group of rational points on the Jacobian variety). In both cases, for the first curve I tried (using the quotient as above) the Mordell-Weil group has rank 1. One can then use Chabauty's method (also implemented in Magma) to determine the set of rational points on the genus 2 curve. In each case, there are three points, but they do not lift to rational points on the given curve.
(EDIT:) Using the quotients via the action on $x$ and $z$ leads to curves whose Jacobians have finite Mordell-Weil groups, leading to a simpler computation.
The three points one finds on the quotient curves correspond to $x = 0$, $y = 0$ and $z = 0$. So if these are the only points there, then one can easily check that there are no rational points on the original curves.