Extremely large-scale multiple-input multiple-output (XL-MIMO) is gaining attention as a prominent technology for enabling the sixth-generation (6G) wireless networks. However, the vast antenna array and the huge bandwidth introduce a non-negligible beam squint effect, causing beams of different frequencies to focus at different locations. One approach to cope with this is to employ true-time-delay lines (TTDs)-based beamforming to control the range and trajectory of near-field beam squint, known as the near-field controllable beam squint (CBS) effect. In this paper, we investigate the user localization in near-field wideband XL-MIMO systems under the beam squint effect and spatial non-stationary properties. Firstly, we derive the expressions for Cramér-Rao Bounds (CRBs) for characterizing the performance of estimating both angle and distance. This analysis aims to assess the potential of leveraging CBS for precise user localization. Secondly, a user localization scheme combining CBS and beam training is proposed. Specifically, we organize multiple subcarriers into groups, directing beams from different groups to distinct angles or distances through the CBS to obtain the estimates of users' angles and distances. Furthermore, we design a user localization scheme based on a convolutional neural network model, namely ConvNeXt. This scheme utilizes the inputs and outputs of the CBS-based scheme to generate high-precision estimates of angle and distance. More importantly, our proposed ConvNeXt-based user localization scheme achieves centimeter-level accuracy in localization estimates.