Scanning Hall probe microscopy has been used to study flux Structures and dynamics in 5 mu m x 5 mu m YBCO thin film squares, which are mesoscopic with respect to the magnetic penetration depth, lambda(T), at temperatures close to T-C. A number of unusual vortex phenomena are observed in these microstructures which differ qualitatively from the expected behaviour of more macroscopic pieces of film. In field-cooled (FC) experiments to similar to 65 K a full Meissner state is generated for cooling fields less than similar to 6 Oe, reflecting the relatively small demagnetization factors in our samples. Cooling in higher fields, however, results in only a very weak diamagnetic response at low temperatures whose magnitude is almost independent of the cooling field. In contrast we observe considerable trapped flux upon field-removal whose magnitude grows monotonically with cooling field. Remarkably, all FC flux distributions exhibit almost perfect rotational symmetry, and can be nearly completely cancelled in a reversible fashion by tuning the field applied to the initially FC state. Our field-cooled and zero-field-cooled results have been analysed in terms of a Bean-like critical state model containing constant edge and bulk current densities, and most of the observed phenomena can be explained by considering the relative weight of these two components. Not all flux profiles call be described by our simple model, however, and under certain circumstances symmetry-breaking 'dipole'-like flux structures can form in several adjacent YBCO squares. We speculate that these are related to the unidirectional At-ion milling process which was used to pattern the squares and could have broken the expected four-fold symmetry. We note that our results could have important implications for the miniaturization of thin film HTS devices.