Recent experiments demonstrated that GaAs/AlAs based micropillar cavities are promising systems for quantum optomechanics, allowing the simultaneous three-dimensional confinement of near-infrared photons and acoustic phonons in the 18-100 GHz range. Here, we investigate through numerical simulations the optomechanical properties of this new platform. We evidence how the Poisson's ratio and semiconductor/vacuum boundary conditions lead to very distinct features in the mechanical and optical three-dimensional confinement. We find a strong dependence of the mechanical quality factor and strain distribution on the micropillar radius, in great contrast to what is predicted and observed in the optical domain. The derived optomechanical coupling constants g0 reach ultra-large values in the 106 rad/s range.