The mechanism of generating backward dragging forces on objects standing in the path of a single, translationally invariant, symmetric acoustic Bessel beam is studied. This paper aims to provide mechanical and structural conditions for the emergence of negative axial forces based on the elastodynamic response of acoustically penetrable objects and beam's nonparaxiality parameter. An extensive numerical study has been performed for various liquid and solid elastic cases to illustrate the validity of the proposed eigenfrequency-based conditions. Results have revealed the existence of a complex but interpretable link between the emergence of negative radiation forces on spheres illuminated by zero-order Bessel beams and the eigenfrequencies of the particle. Considerable progress has been made in elucidating these relationships, which may lead to the development of predictable and robust single-beam acoustic handling devices.