Yes definitely use the mesh grids. I recently tried building a cage of the regular grids for my new ferret rescues and just like the chinchillas, they will stick their head through the grid holes and can't pull them back out. I had to use wire snips to free one of the ferrets heads, so now I'm dismantling the entire thing. Double-gridding doesn't look so neat to me so I went ahead and bought them a huge ferret cage online.
In my opinion, these C&C cages (built of regular grids) are only safe for guinea pigs.