With first principles calculations, we predict a novel stable 2D layered structure for group VI elements Se and Te that we call square selenene and square tellurene, respectively. They have chair-like buckled structures similar to other layered materials such as silicene and germanene but with a square unit cell rather than a hexagonal one. This special structure gives rise to anisotropic band dispersions near the Fermi level that can be described by a generalized semi-Dirac Hamiltonian. We show that the considerably large band gap (~0.1 eV) opened by spin-orbit coupling makes square selenene and tellurene topological insulators, hosting non-trivial edge states. Therefore, square selenene and tellurene are promising materials for novel electronic and spintronic applications. Finally, we show that this new type of 2D elemental materials can potentially be grown on proper substrates, such as a Au(1 0 0) surface.