This paper numerically investigates the impact of fluid rheology on the behaviors of a spherical capsule through a microchannel constriction. Different flow scenarios are considered: a Newtonian capsule in a viscoelastic matrix, a Newtonian capsule in a Newtonian matrix, and a viscoelastic capsule in a Newtonian matrix. The results demonstrate that the capsule's lengths undergo oscillations during the passage through the constriction, with three stages of evolution. When approaching the constriction, the capsule respectively experiences increase and decrease in its length and height. While within or exiting the constriction, the length of the capsule continuously decreases, and the height generally increases. As the capsule moves away from the constriction, the capsule relaxes to different profiles in different flows. Detailed analysis on the effects of the fluid viscoelasticity on the capsule's lengths in different stages is provided. In addition, the behaviors of a red blood cell passing through a microchannel constriction are also examined. This study sheds light on the complex behaviors of a spherical capsule and red blood cell in microchannel constriction, emphasizing the significant influence of fluid rheology on their deformation and shape changes.