The ball you forced inside is affecting the magnetic field.
Note that each of these magnets has a north and south pole--the poles are hemispherical. Also, these poled are perfectly aligned in the cylinder.
In the first case, there are two forced acting here:
The force of attraction between neighboring magnets--this will resist the deformation and will try to get the cylinder back to its original state. This is because there are regions formed with a not-so-perfect attraction.
Force of attraction and repulsion between opposing elements. In certain cases, the opposing magnets will repel, and they will attract in other cases. So there will be a grid of attraction and repulsion formed when you squeeze the magnet, and the net effect will be zero. There will only be perfect attraction is you manage to skew it so that the "top" and "bottom" layers are hexagonally packed as well--pretty hard since there will be repulsion elsewhere. Most probably the resistance to squeezing manifests itself around halfway, and increases from there.
When you add a magnetizable ball, it aligns itself at a spot and becomes half a magnet as well.
Note that I've added it on the outside in the diagram, since I can't figure out how to add it to draw it on the inside. When the ball is inside, the same thing happens.
Now, this makes it much easier for the opposing sides to touch each other
The point is, even though the magnetized ball isn't that good a magnet, its still touching the sides. And this means that the magnetic attraction is large. On the other hand, whatever deformation repulsion is there is still pretty small, since you haven't deformed it that much yet. So, the attractive force wins, and it stays in place.