Optical time-stretch (OTS) imaging flow cytometry offers a promising solution for high-throughput and high-precision cell analysis due to its capabilities of high-speed, high-quality, and continuous imaging. Compressed sensing (CS) makes it practically applicable by significantly reducing the data volume while maintaining its high-speed and high-quality imaging properties. To enrich the information of the images acquired with CS-equipped OTS imaging flow cytometry, in this work we propose and experimentally demonstrate Fourier-domain-compressed OTS quantitative phase imaging flow cytometry. It is capable of acquiring intensity and quantitative phase images of cells simultaneously from the compressed data. To evaluate the performance of our method, static microparticles and a corn root cross section are experimentally measured under various compression ratios. Furthermore, to show how our method can be applied in practice, we utilize it in the drug response analysis of breast cancer cells. Experimental results show that our method can acquire high-quality intensity and quantitative phase images of flowing cells at a flowing speed of 1 m/s and a compression ratio of 30%. Combined with machine-learning-based image analysis, it can distinguish drug-treated and drug-untreated cells with an accuracy of over 95%. We believe our method can facilitate cell analysis in both scientific research and clinical settings where both high-throughput and high-content cell analysis is required.