A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper. We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety.