In this paper, we focus our attention on the mitigation of beam squint for reconfigurable intelligent surface (RIS) aided wideband millimeter wave (mmWave) communications. Due to the intrinsic passive property, the phase shifts of all elements in RIS should be the same for all frequencies. However, in the wideband scenario, beam squint induced distinct path phases require designing different phase shifts for different frequencies. The above irreconcilable contradiction will dramatically affect the system performance, considering the RIS usually consists of enormous elements and the bandwidth of wideband mmWave communications may be up to several GHz. Therefore, we propose some novel phase shift design schemes for mitigating the effect of beam squint for both line-of-sight (LoS) and non-Los (NLoS) scenarios. Specifically, for the LoS scenario, we firstly derive the optimal phase shift for each frequency and obtain the common phase shift by maximizing the upper bound of achievable rate. Then, for the NLoS scenario, a mean channel covariance matrix (MCCM) based scheme is proposed by fully exploiting the correlations between both the paths and the subcarriers. Our extensive numerical experiments confirm the effectiveness of the proposed phase shift design schemes.