The properties of an atomic lubricant confined between two approaching solids are investigated by a model that accounts for the curvature and elastic properties of the solid surfaces. Well defined atomic layers develop in the lubricant film when the width of the film is of the order of a few atomic diameters. An external squeezing-pressure induces discontinuous, thermally activated changes in the number n of lubricant layers. The precise mechanism for these layering transitions depends on n, and on the lubricant-surface pinning barriers. Thus, in the absence of sliding, unpinned or weakly pinned incommensurate lubricant layers give rise to fast and complete layering transitions. Strongly pinned incommensurate and commensurate layers give rise to sluggish and incomplete transformations, resulting in trapped islands. In particular, for commensurate layers it is often not possible to squeeze out the last few lubricant layers. However, lateral sliding of the two solid surfaces breaks down the pinned structures, greatly enhancing the rate of the layering transitions. In the case of sliding, an important parameter is the barrier for sliding one lubricant layer with respect to the others. When this barrier is larger than the lubricant-surface pinning barrier, the lubricant film tends to move like a rigid body with respect to the solid surface. In the opposite case, slip events may occur both within the lubricant film and at the lubricant-solid interface, making the squeeze-out process much more complex. In some of the simulations we observe an intermediate phase, forming immediately before the layering transition. This transient structure has a lower 2D density than the initial phase, and allows the system to release elastic energy, which is the driving force for the phase transformation.