In this paper, we investigate channel estimation for reconfigurable intelligent surface (RIS) empowered millimeter-wave (mmWave) multi-user single-input multiple-output communication systems using low-resolution quantization. Due to the high cost and power consumption of analog-to-digital converters (ADCs) in large antenna arrays and for wide signal bandwidths, designing mmWave systems with low-resolution ADCs is beneficial. To tackle this issue, we propose a channel estimation design using task-based quantization that considers the underlying hybrid analog and digital architecture in order to improve the system performance under finite bit-resolution constraints. Our goal is to accomplish a channel estimation task that minimizes the mean squared error distortion between the true and estimated channel. We develop two types of channel estimators: a cascaded channel estimator for an RIS with purely passive elements, and an estimator for the separate RIS-related channels that leverages additional information from a few semi-passive elements at the RIS capable of processing the received signals with radio frequency chains. Numerical results demonstrate that the proposed channel estimation designs exploiting task-based quantization outperform purely digital methods and can effectively approach the performance of a system with unlimited resolution ADCs. Furthermore, the proposed channel estimators are shown to be superior to baselines with small training overhead.