In this study, we present a detailed analysis of the squeeze-out dynamics of nanoconfined water confined between two hydrophilic surfaces measured by small-amplitude dynamic atomic force microscopy (AFM). Explicitly considering the instantaneous tip-surface separation during squeezeout, we confirm the existence of an adsorbed molecular water layer on mica and at least two hydration layers. We also confirm the previous observation of a sharp transition in the viscoelastic response of the nanoconfined water as the compression rate is increased beyond a critical value (previously determined to be about 0.8 nm/s). We find that below the critical value, the tip passes smoothly through the molecular layers of the film, while above the critical speed, the tip encounters "pinning" at separations where the film is able to temporarily order. Pre-ordering of the film is accompanied by increased force fluctuations, which lead to increased damping preceding a peak in the film stiffness once ordering is completed. We analyze the data using both Kelvin-Voigt and Maxwell viscoelastic models. This provides a complementary picture of the viscoelastic response of the confined water film.