In this work, we have developed a facile, economical microfluidic approach as well as a simple model description to measure and predict the mechanical properties of composite core-shell microparticles made from materials with dramatically different elastic properties. By forcing the particles through a tapered capillary and analyzing their deformation, the shear and compressive moduli can be measured in one single experiment. We have also formulated theoretical models that accurately capture the moduli of the microparticles in both the elastic and the non-linear deformation regimes. Our results show how the moduli of these core-shell structures depend on the material composition of the core-shell microparticles, as well as on their microstructures. The proposed technique and the understanding enabled by it also provide valuable insights into the mechanical behavior of analogous biomaterials, such as liposomes and cells.