This paper proposes a novel reconfigurable intelligent surface (RIS)-assisted generalized receive quadrature spatial modulation (RIS-GRQSM) scheme to enhance the spectral efficiency (SE) of RIS-aided \textit{quadrature} spatial modulation (QSM) systems. By leveraging the principle of \textit{generalized} spatial modulation (GSM), multiple receive antennas are independently activated for \textit{both} the in-phase and quadrature components of spatial symbols. To fully exploit the potential of RIS, we formulate a max-min optimization problem to adjust the phase shifts of all RIS elements, thereby maximizing the effective signal-to-noise ratios (SNRs) at the activated antennas. Using Lagrange duality, the original high-dimensional non-convex problem is reduced to a tractable problem with a smaller number of real variables, and a closed-form suboptimal solution is also proposed, which achieves near-optimal performance with a sufficiently large RIS. At the receiver, a low-complexity non-coherent energy-based greedy detector (GD) is introduced for efficient symbol detection. We further extend the RIS-GRQSM framework to a multicast communication system, where all users receive identical information with equal SNR levels, and provide a detailed performance analysis of both systems. In particular, we derive the average bit error probability (ABEP) for the proposed RIS-GRQSM and multicast systems under optimal and suboptimal optimization strategies. Numerical results show that RIS-GRQSM significantly improves the SE and error rate performance compared with benchmark schemes, while the multicast extension achieves performance close to benchmark methods at substantially lower complexity.