Inspired by the recent experiments of cavity-magnon system and the theories of nonlinear reservoir, a scheme of coupling a cavity-magnon system to a squeezed reservoir is proposed to analyze the magnon blockade effect, thus realizing the manipulation of magnetism at quantum level. An yttrium iron garnet sphere was located in a driven microwave cavity. The Kittel mode in the yttrium iron garnet sphere interacted with the microwave cavity mode, and the cavity field was coupled to a squeezed vacuum reservoir. By numerically solving the quantum master equation of the system, the influence of various factors such as coupling strength, detuning and dissipation rate on the correlation function was analyzed in details. It is proved theoretically that the cavity-magnon system coupled to a squeezed reservoir can induce single-magnon blockade and two-magnon blockade, and the switching among single-magnon blockade, two-magnon blockade and two-magnon tunneling can be flexibly controlled by tuning the driving intensity or the detuning. The magnon blockade effect in the proposed scheme is mainly induced by the nonlinearity of the squeezed reservoir, which provides a possible new method to realize single or double magnon blockade in cavity-magnon system.