My chin does the same thing!! He pushes his bowl (with food in it) off his shelf and it lands upside down and hay gets everywhere! 
For safety, bowls or anything that has any weight to it at all, should be attached securely to the side of the cage or placed on the floor of the cage. Falling objects can break bones and/or kill ... remember chinchilla bones are about the same size as a toothpick, some even smaller.
"J" feeders are good and can be attached on the inside of the cage or, on the outside with the serving part inside. They are much less likely to be urinated in also. Most have a screen bottom and allow the crumbs to fall through the bottom of the inside of the cage or on the shelf.
The shiny metal bowls you find in the bird department, have holders that can be attached to the side of the cage and the bowls are easily removed to wash ... again, I think it is best, to mount these on the lowlest level of the cage or low enough on the shelf that a chin can not get under it ... I learned the hard way ... I had a couple of chins that liked to get under the bowls and push them upward and out of their holder ... spilling all the food everywhere.
For safety, food bowls should be ceramic or something heavy and have straight sides with the base the same size or larger than the top ... this prevents them from tipping it over in most cases. But ... be cautious ... the smaller feeder bowls that do not weigh very much can be tipped over, should a chin choose to sit on the top rim ... this could be dangerous for the chinchilla, but even more dangerous for any kits in the cage, the bowl itself is heavy enough to kill a kit or young chinchilla. One of my momma chins was having a fit one day, I could not figure out what was wrong, other than her small ceramic food bowl was upside down ... when I picked it up, there was one of the kits she had delivered the day before, it was OK, but would not have been for very long ... it could have suffocated in a short period of time. I changed to "J" feeders and larger ceramic bowls that could not easily be tipped over within a week's time.
...my chins eat about half of the cubes! Maybe the good stuff is on the outside ...
Cubes or loose hay that is left on a shelf or the bottom of the cage for more than a few hours, is often urinated on, accidentally or other wise ... the chin will not eat it once this happens ... plus, it provides a perfect atmosphere for bacteria and/or mold to grow. It is sometimes best to give smaller portions twice a day.
Always remove any hay that has been in the cage for 24 hours, or if it feels damp or moist.
Jo Ann
