Bing's house deals with the analogous question of collapsibility. It is an example of a contractible non-collapsible space. For that problem it is known that if you allow both collapses and anti-collapses, every contractable space can be reduced to a point. This follows from "Simple homotopy theory". I dont know if this result of simple homolopy theory extends to animals built from cubes but they might.
Gil Kalai
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