A novel movable-element simultaneously transmitting and reflecting surface (ME-STARS)-assisted near-field wideband communication framework is proposed. In particular, the position of each STARS element can be adjusted to combat the significant wideband beam squint issue in the near field instead of using costly true-time delay components. Four practical ME-STARS element movement modes are proposed, namely region-based (RB), horizontal-based (HB), vertical-based (VB), and diagonal-based (DB) modes. Based on this, a near-field wideband multi-user downlink communication scenario is considered, where a sum rate maximization problem is formulated by jointly optimizing the base station (BS) precoding, ME-STARS beamforming, and element positions. To solve this intractable problem, a two-layer algorithm is developed. For the inner layer, the block coordinate descent optimization framework is utilized to solve the BS precoding and ME-STARS beamforming in an iterative manner. For the outer layer, the particle swarm optimization-based heuristic search method is employed to determine the desired element positions. Numerical results show that:1) the ME-STARSs can effectively address the beam squint for near-field wideband communications compared to conventional STARSs with fixed element positions; 2) the RB mode achieves the most efficient beam squint effect mitigation, while the DB mode achieves the best trade-off between performance gain and hardware overhead; and 3) an increase in the number of ME-STARS elements or BS subcarriers substantially improves the system performance.