If $X$ is a curve, then just take linear projection from a disjoint codimension 2 linear subspace in the spanning hyperplane of $Z$. If the dimension of $X$ is $2$ or more, then there never exists such an étale neighborhood and morphism. For $Z$ of dimension $n-1 \geq 1$, for the normal sheaf $\mathcal{N}_{Z/X}$, the top intersection number $c_1(\mathcal{N}_{Z/X} )^{n-1}$ equals the degree of $X$, which is positive. Since $\mathcal{N}_{Z/X}$ equals $\mathcal{N}_{Z/U}$, also $c_1(\mathcal{N}_{Z/U})^{n-1}$ is positive. However, for a fiber of a morphism, $\mathcal{N}_{Z/U}$ is isomorphic to $\mathcal{O}_Z$ so that $c_1(\mathcal{N}_{Z/U})$ equals $0$.