Beyond diagonal reconfigurable intelligent surface (BD-RIS), with its enhanced degrees of freedom compared to conventional RIS, has demonstrated notable potential for enhancing wireless communication performance. However, a key challenge in employing BD-RIS lies in accurately acquiring its channel state information (CSI) with both the base station (BS) and users. Existing BD-RIS channel estimation methods rely mainly on dedicated pilot signals, which increase system overhead and may be incompatible with current communication protocols. To overcome these limitations, this letter proposes a new single-layer neural network (NN)-enabled channel estimation method utilizing only the easily accessible received power measurements at user terminals. In particular, we show that the received signal power can be expressed in a form similar to a single-layer NN, where the weights represent the BD-RIS's CSI. This structure enables the recovery of CSI using the backward propagation, based on power measurements collected under varying training reflection coefficients. Numerical results show that our proposed method can achieve a small normalized mean square error (NMSE), particularly when the number of training reflections is large.