Based on one dimensional high-order nonlinear wave model, the possibility and existence condition of kink and anti-kink solitary waves in microstructured solids are analyzed with the qualitative analysis method of dynamic system. Analysis results show that the kink and anti-kink solitary waves can exist in high-order nonlinear and anomalous dispersion microstructured solids under appropriate conditions, and the existence conditions are given exactly. Influences of high-order microscale nonlinear effect on the kink and anti-kink solitary waves are analyzed with the numerical method. Results show that the kink and anti-kink solitary waves become more and more steep (or gentle) with the negative reinforcing (or positive reinforcing) of high-order microscale nonlinear effect, while their amplitudes remain unchanged.