Those of you who aren't familiar with the concept of Non-Newtonian Fluids, the thing to realize is that they are fluid or solid depending on the stress exerted on them. If you fill a swimming pool with a non-Newtonian fluid, you'll be able to walk right across it. So long as you remain in motion, it will support your weight. As soon as you stop, however, you'll sink. It's a bizarre property of matter that's usually outside of our own everyday experience.
And so a non-Newtonian fluid will act solid or liquid depending on the forces exerted on it.
You can see a good demonstration in the video below:
Could you use it as a cushion in the same way foam rubber sheeting is used in Judo mats? I don't think it has ever been done, but I can guess as to why that may be. If you think about it, you don't want something that feels solid when you're impacting on it. You want to feel some give. A non-Newtonian fluid would react to a falling judoka by feeling as hard as a brick at first. Then a few seconds later, it will feel squishy and will displace around the judoka.
So for padding impacts, it fails.
Also, if you're just walking across a hypothetical mat filled with a non-Newtoninan fluid, it's going to be fine. You'll feel like there's solid ground under you. But as soon as you stop, your feet will squish in to the mat, leaving your feet in a 2 inch depression. That's a tripping hazard and a good way to stub your toe.
Lastly, foam rubber is dry. If you puncture or rip that judo mat, the worst that happens is you expose some of the foam rubber material inside. There are vinyl repair kits that easily seal that hole. But if it's filled with a non-Newtonian fluid, it will just leak right out with the first tiny puncture.
So to answer the question, no, I don't believe this could be used in judo mats very effectively. You could try to do a kind of hybrid material of some sort which combines a foam scaffolding with some kind of non-Newtonian fluid. But it's hard to see how that would be an improvement over what already exists. And you would still have problems with leaks.
Hope that helps.