Cellular mechanical phenotypes in connection to physiological and pathological states of cells have become a promising intrinsic biomarker for label-free cell analysis in various biological research and medical diagnostics. In this work, we present a microfluidic system capable of high-throughput cellular mechanical phenotyping based on a rapid single-cell hydrodynamic stretching in a continuous viscoelastic fluid flow. Randomly introduced single cells are first aligned into a single streamline in viscoelastic fluids before being guided to a flow splitting junction for consistent hydrodynamic stretching. The arrival of individual cells prior to the flow splitting junction can be detected by an electrical sensing unit, which produces a triggering signal to activate a high-speed camera for on-demand imaging of the cell motion and deformation through the flow splitting junction. Cellular mechanical phenotypes, including cell size and cell deformability, are extracted from the analysis of these captured single-cell images. We have evaluated the sensitivity of the developed microfluidic mechanical phenotyping system by measuring the synthesized hydrogel microbeads with known Young's modulus. With this microfluidic cellular mechanical phenotyping system, we have revealed the statistical difference in the deformability of microfilament disrupted, normal, and fixed NIH 3T3 fibroblast cells. Furthermore, with the implementation of a machine-learning-based classification of MCF-10A and MDA-MB-231 mixtures, our label-free cellular phenotyping system has achieved a comparable cell analysis accuracy (0.9:1, 5.03:1) with respect to the fluorescence-based flow cytometry results (0.97:1, 5.33:1). The presented microfluidic mechanical phenotyping technique will open new avenues for high-throughput and label-free single-cell analysis in diverse biomedical applications.