We experimentally evaluated the accuracy with which material properties can be estimated through object compression by two standard parallel jaw grippers and a force/torque sensor mounted at the robot wrist, with a professional biaxial compression device used as reference. Gripper effort versus position curves were obtained and transformed into stress/strain curves. The modulus of elasticity was estimated at different strain points and the effect of multiple compression cycles (precycling), compression speed, and the gripper surface area on estimation was studied. Viscoelasticity was estimated using the energy absorbed in a compression/decompression cycle, the Kelvin-Voigt, and Hunt-Crossley models. We found that: (1) slower compression speeds improved elasticity estimation, while precycling or surface area did not; (2) the robot grippers, even after calibration, were found to have a limited capability of delivering accurate estimates of absolute values of Young's modulus and viscoelasticity; (3) relative ordering of material characteristics was largely consistent across different grippers; (4) despite the nonlinear characteristics of deformable objects, fitting linear stress/strain approximations led to more stable results than local estimates of Young's modulus; (5) the Hunt-Crossley model worked best to estimate viscoelasticity, from a single object compression. A two-dimensional space formed by elasticity and viscoelasticity estimates obtained from a single grasp is advantageous for the discrimination of the object material properties. We demonstrated the applicability of our findings in a mock single stream recycling scenario, where plastic, paper, and metal objects were correctly separated from a single grasp, even when compressed at different locations on the object. The data and code are publicly available.