In this work, we propose a flexible block coordinate method for unconstrained optimization problems under Hölder continuity assumptions. The method guarantees convergence to stationary points and has worst-case complexity results comparable to those obtained by single-block methods that assume Lipschitz or Hölder continuity. The approach is based on quadratic models of the objective function combined with quadratic regularization. Flexible block selection strategies are allowed, and different sufficient descent conditions previously considered in the literature are unified within the same framework. The proposed method is fully implementable and incorporates a practical certificate of stationarity as a stopping criterion. Well-definiteness and convergence are established under the Hölder continuity of the objective function gradient, extending classical results that typically rely on Lipschitz assumptions. Illustrative numerical tests are performed and a freely available Julia implementation is provided.