The transport of deformable objects, including polymer particles, vesicles, and cells, has been a subject of interest for several decades where the majority of experimental and theoretical studies have been focused on circular tubes. Due to advances in microfluidics, there is a need to study the transport of individual deformable particles in rectangular microchannels where corner flows can be important. In this study, we report measurements of hydrodynamic mobility of confined polymeric particles, vesicles, and cancer cells in a linear microchannel with a square cross-section. Our operating conditions are such that the mobility is measured as a function of geometric confinement over the range 0.3 < λ < 1.5 and at specified particle Reynolds numbers that are within 0.1 < Rep < 2.5. The experimental mobility data of each of these systems is compared with the circular-tube theory of Hestroni, Haber, and Wacholder [J. Fluid Mech. 41, 689-705 (1970)] with modifications made for a square cross-section. For polymeric particles, we find that the mobility data agrees well over a large confinement range with the theory but under predicts for vesicles. The mobility of vesicles is higher in a square channel than in a circular tube, and does not depend significantly on membrane mechanical properties. The mobility of cancer cells is in good agreement with the theory up to λ ≈ 0.8, after which it deviates. Comparison of the mobility data of the three systems reveals that cancer cells have higher mobility than rigid particles but lower than vesicles, suggesting that the cell membrane frictional properties are in between a solid-like interface and a fluid bilayer. We explain further the differences in the mobility of the three systems by considering their shape deformation and surface flow on the interface. The results of this study may find potential applications in drug delivery and biomedical diagnostics.