There are no nontrivial chordal square-complementary graphs

We study square-complementary graphs $G$ (satisfying $G^2\cong \bar{G}$). We show that in such graphs, no two vertices have comparable closed neighborhoods. This implies that nontrivial square-complementary graphs have no simplicial vertices and are not chordal, thus solving two open problems posed in Discrete Mathematics 327 (2014) 62-75. We also show that no nontrivial square-complementary graph is distance-hereditary.