We investigate the zero-temperature phase diagram of the fully frustrated transverse-field Ising model on the square lattice both in the classical limit and in the presence of quantum fluctuations. At the classical level (the limit of infinite spin S), we find that upon decreasing the transverse field Gamma this model exhibits a phase transition from the fully polarized state into an eightfold degenerate translational symmetry breaking state. This phase can be identified to correspond to plaquette order in the dimer language and remains the lowest-energy state in the entire range of fields below the critical one, Gamma(c). The eightfold degenerate solution that corresponds to columnar order in the dimer language is a saddle point of the classical energy. It is degenerate with the plaquette solution at Gamma = 0 and is only slightly higher in energy in the whole interval 0 < Gamma < Gamma(c). The effect of quantum fluctuations is investigated in the context of a large S expansion both for the plaquette and columnar structures. For this purpose we employ an approximate method allowing to estimate from above the fluctuation-induced correction to the energy of a configuration which at the classical level is a saddle point of the energy, not a local minimum, and find that the harmonic quantum fluctuations show a clear tendency to overcome the energy difference between the two states. For relatively high fields, the transition from the plaquette to the columnar state takes place at values of S large enough to be in the domain of validity of the harmonic approximation.