I wonder if massive bodies at the "edges" of the universe aren't being 'squirted" outwards, so to speak.

If a massive bodiy were to be somehow hurled into the universe, as it moved toward the centre of the universe, all of the pre-existing bodies in the universe would move slightly towards it by gravitational attraction. So it therefore follows that as the massive bodies in the central region of the universe draw together by gravitational attraction, that the massive bodies at the 'edges' would correspondingly move outwards.

Is that the "pushing" form of gravity to which you allude?