From my observation (I have six BA's) the soft rubber shell and dropping the eggs from the roost go together. When the shells get harder, they lay the eggs in the nest with the most golf balls.
The paper thin and shellless eggs are a thing of the past (at least for now) at my house. I can think of four changes:
1) put oyster shell in the food and out for free choice (not every day anymore, but every so often)
2) started putting ACV (apple cider vinegar) in the water.
3) switched from pellets to crumbles and put more food out (I'd leave the food out free choice, but the wild doves get in it)
4) They got a month older.
I have 2 degrees in science. I have no idea why the ACV works, but I am convinced that it helps. I don't use the organic stuff with the mother in it that everyone swears by. I probably would, but I sent my mom to the store the first time and she came home with regular distilled brown ACV. I was feeling desperate so started it right away. Out here in the desert, I have at least 3 waterers at all times. I usually only have the ACV in one - in case they need a break from it. It needs to go in a plastic one otherwise it corrodes the metal off, not good for the chickens. I use 1T per gallon and at first put it in the water located near their afternoon hangout. In 110 plus, they stayed there during the hottest part of the day and I am sure drank a good dose from it.
Now, I started the ACV due to a lethargic sick hen. (I was very, very worried. ) However, I think it helped with the eggs too.
As previously mentioned, I also switched to layer crumbles and put more out. They like it better and ate it with more gusto. I was concerned they weren't eating enough of the layer food to get needed nutrients and wanted to make sure they got all they needed.
I don't really know which thing made the difference and once in a while I get soft egg or a white one, but not like it used to be with shelless ones or paper thin ones dropped from the roost every night.
Another possibility, I suppose, is that the young layers got older and better at laying...but I think that is only a small factor.