Is it possible you have an egg eater? That barred rock looks pretty ready to lay, especially if that photo was taken a month ago.
What makes me ask is, if they have been kicking out the wooden eggs, it would be very easy for a first real egg to have got kicked out and smashed and they tasted it and liked it and then learned that it's worth trying to break the eggs. They can clean up a broken egg pretty fast and effectively. Another contributary factor that might lead to this is that their run isn't very big, so they might be more inclined to look for mischief in the nest boxes out of boredom. Do the nest boxes have a good high lip on the front? If not, I would amend that, so it's much harder for them to kick the nesting material and eggs out. Also, if you can, perhaps give them a larger run by building an add on extension or consider giving them toys (a simple small plastic drinks bottle with a little hole drilled in the side filled with pellets or meal worms or scratch that they have to push/kick around to get the contents to come out and maybe make a little swing or put simple perches in it and scatter their feed in a scratching area filled with wood chip or dead leaves etc, so they have to work for their food rather than eat from a feeder and then look for mischief the rest of the day. Don't worry too much about waste as they can clean up a small area pretty quickly and effectively. I have a feeder with layer pellets in the hen house but I scatter mixed corn in the paddock and yard twice a day for mine and they spend time picking that up and then scratching in case they missed any and then they move into the stables/muck heap and go through the horses bedding and then they have a bit of a doze or go and lay eggs or dust bath. Do yours have a dust bath area. If not create one for them. It's all part of having healthy happy chickens.
I can't advise on brand of feed I'm afraid as I don't think you will have the same as here in the UK. I was just using the cheapest layer pellets here through the summer which was fine but then since some of the older ladies were going through a hard moult I wanted to up the protein level for them, hence changing to the one I am currently using, but I plan to change back very soon, now they are mostly over the moult.
Hope that info has given you some food for thought.
Best wishes
Barbara