We consider the optimal stopping problem with non-linear f-expectation (induced by a BSDE) without making any regularity assumptions on the pay-off process $\xi$. We show that the value family can be aggregated by an optional process Y. We characterize the process Y as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with $\xi$ as the obstacle. This characterization is established by first showing existence and uniqueness for the Reflected BSDE with irregular obstacle and also a comparison theorem.