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We propose quantum receivers with optical squeezing and photon-number-resolving detectors (PNRDs) for the near-optimal discrimination of quaternary phase-shift-keyed coherent state signals. The basic scheme is similar to the previous proposals [e.g., Izumi et al., Phys. Rev. A 86, 042328 (2012)] in which displacement operations, on-off detectors, and electrical feedforward operations were used. Here we study two types of receivers, one of which installs optical squeezings and the other uses PNRDs instead of on-off detectors. We show that both receivers can attain lower error rates than that in the previous scheme. In particular, we show the PNRD-based receiver has a significant gain when the ratio between the mean photon number of the signal and the number of the feedforward steps is relatively high, in other words, when the probability of detecting two or more photons at each detector is not negligible. Moreover, we show that the PNRD-based receiver can suppress the errors due to dark counts, which the receiver with the on-off detector cannot do with a small number of feedforwards.