Non-diagonal reconfigurable intelligent surfaces (RISs) offer enhanced wireless signal manipulation over conventional RIS by enabling the incident signal on any of its $M$ elements to be reflected from another element via an $M \times M$ switch array. To fully exploit this flexible configuration, the acquisition of individual channel state information (CSI) is essential. However, due to the passive nature of the RIS, cascaded channel estimation is performed, as the RIS itself lacks signal processing capabilities. This entails estimating the CSI for the $M \times M$ switch array cascaded channels, resulting in estimating $M^2$ coefficients, to identify the optimal configuration of the non-diagonal RIS that maximizes the channel gain. In this paper, we propose a low-complexity channel estimation protocol that substantially reduces the estimation overhead from the exhaustive $M^2$ coefficients to only $3M$ coefficients for both single-input single-output (SISO) and multiple-input single-output (MISO) systems. Specifically, a three-stage pilot-based protocol is proposed to estimate scaled versions of the user-RIS and RIS-base-station (BS) channels in the first two stages using the least square (LS) estimator and the commonly used ON/OFF protocol for conventional RIS. These scaled estimates enable the optimization of the switch array. In the third stage, the cascaded user-RIS-BS channels are estimated to enable efficient beamforming optimization. Complexity analysis shows that our proposed protocol significantly reduces the BS estimation complexity from $\mathcal{O}(NM^2)$ to $\mathcal{O}(NM)$, where $N$ is the number of BS antennas. This complexity is similar to the ON/OFF-based LS estimation for conventional diagonal RIS.