More trouble than it's worth. The current style of lightweight rubber or plastic bumper is already elastic for small collisions and absorbs some energy from larger ones, deforming as it goes. Replacing that with a comparitively complex sac of non-newtonian fluid will be more expensive to manufacture, more liable to vandalism, heavier, harder to repair, so on and so forth.

It's also potentially able to make a big mess from a small accident that a regular bumper would shake off. Jam an iron bar into a regular bumper and you'll scuff it. Jam it into a non-newtonian sac, and you might rupture it.