Question about image equalization
It probably has the potential to do so, but the probability is marginal. Result of equalization depends on the initial pixel intensity distribution.

Image histogram equalization can have different implementations.
In one possible implementation, you would re-map a whole block of a given intensity level into a new calculated intensity. In such a case, objects will not change shape; boundaries will remain the same.

The aim of equalization is not necessarily the creation of a new image with strictly flat histogram will all bins equally occupied. That won't create a good looking image in any sense. Instead, the aim is to expand the pixel intensities to fully utilize the available dynamic range in the hope of getting a higher constrast with more visible dark regions and unsaturated highlights.