The radial eigenmodes of geodesic acoustic modes (GAMs) in tokamaks are investigated in the presence of equilibrium toroidal rotation flow (ETRF) using the gyro-kinetic formalism. The radial differential equation for finite-Larmor-radius effects induced GAM eigenmodes is constructed from the local dispersion relation. In the non-rotation case, we analyze the radial structures and the number of GAM eigenmodes for different magnetic shear q-profiles and compare the results with previous theoretical and experimental studies. We find that the number of GAM eigenmodes exhibits only weak sensitivity to the q-profile, which differs from previous two-fluid theory predicting that a reversed shear q-profile may accommodate more GAM eigenmodes than a normal shear q-profile (Wang et al 2021 Nucl. Fusion61 106024). Previous experimental observations and theoretical studies have shown that the amplitude of GAM eigenmodes peaks near the radial position where the eigenfrequency equals the local continuum GAM frequency. Our results further elucidate this feature by demonstrating that the mode structure can be modulated by the radial variation of collisionless Landau damping, with low-q regions tending to weaken the mode amplitude. With toroidal rotation, the eigenfrequencies increase with the Mach number, consistent with the upward shift of the local GAM continuum. For GAM eigenmodes near the edge, increasing the toroidal Mach number produces a squeezing effect to the oscillatory domain. This behavior is explained by the correlation with the local continuum. The squeezing effect moves the mode outward due to the outward shift of the continuum intersection point.