We analyse the effects of an ambient yield-stress fluid on the flow around squirming microswimmers, employing detailed computational results alongside asymptotic methods to evaluate the swimming speed. We show that a yield stress reduces the swimming speed of spheroidal squirmers and limits the disturbance produced by the swimmer to a finite envelope around itself. A relatively weak yield stress is a singular perturbation to the Newtonian case, for which we employ matched asymptotic expansions to determine a universal, quasi-analytical prediction for the reduction in swimming speed and the dimension of the disturbed region around the swimmer. With a relatively strong yield stress, we find there are distinct behaviours for thin and wide spheroids, delineated by a critical aspect ratio which we determine analytically. For sufficiently wide swimmers the disturbed region is localised in boundary layers around the swimmer and the force-free swimming speed decays with increasing yield stress. In contrast, for thin swimmers the size of the disturbed region remains on the scale of the swimmer and the swimming speed asymptotes to a finite value with increasing yield stress.