Based on Biot-Tsiklauri model and the combined Biot/squirt model, a universal acoustic model for non-Newtonian fluid (Maxwell fluid) saturated porous medium with an arbitrary pore size distribution is presented, in which the squirt-flow mechanism has been taken into account. The influences of non-Newtonian effect of Maxwell fluid on the attenuation and the dispersion characteristics of elastic waves propagating in such a porous medium are investigated. It shows that the non-Newtonian effect and the squirt-flow effect are the important factors for elastic wave dispersion and attenuation. When the squirt flow mechanism occurs, the squirt flow mechanism dominates the contribution to the energy loss of compressional waves at low frequencies. It shows that the squirt flow only affects the dispersion and the attenuation characteristics of two compressional waves, while the non-Newtonian effect of Maxwell fluid not only affects the dispersion and the attenuation characteristics of the compressional waves, but also influences the dispersion and the attenuation characteristics of shear waves.