The finite difference method (FDM) is traditionally restricted to structured grids, limiting its applicability to complex geometries. In this paper, we propose a flexible-node finite difference method (FN-FDM) based on the discrete equivalence equation and its discrete-rule (DEER) framework. FN-FDM enables FDM to handle both ordered and disordered nodes within a unified framework by adaptively constructing computational stencil nodes from arbitrarily distributed neighbors, eliminating the need for ordered data structures. A gradient-based interpolation scheme is employed to achieve second-order spatial accuracy. Numerical experiments, including freestream preservation, isentropic vortex, supersonic flow over dual ellipses, and shock interactions around a hemisphere, demonstrate that FN-FDM preserves vortical structures, captures shocks effectively, and adapts seamlessly to overset grids without interpolation-induced errors. This method significantly reduces the difficulty of generating body-fitted grids and offers a new approach for automated grid generation and computational fluid dynamics (CFD) simulations on complex geometries.