Quiver Grassmannians of equioriented type $\texttt{A}$ and nilpotent equioriented type $\tilde{\texttt{A}}$ quiver representations are GKM-varieties. In particular, they have a cellular decomposition and admit a torus action with finitely many fixed points and one-dimensional orbits (i.e. skeletal action). We examine the case of string representations and provide a classification of all corresponding quiver Grassmannians with a GKM-variety structure.