Initially it was just one egg but then we started to get more, sometimes even several/day. So we began to research what could be causing this.
What we saw as the most common culprit was not enough calcium in the birds' diet. We do feed our girls a premium layer feed, which says it shouldn't require any supplemental cadmium. However, this obviously isn't cutting it. I went to our local feed store and found a 50-lb bag of ground up oyster shells. We gave them a quantity of this stuff and they immediately started to to pick at it. The pieces seem big but I guess the idea is that they eat it and it sits in their guts for a while, releasing calcium.
We also read that vitamin d is important for helping them absorb the calcium, just like it is for us. Since our run is covered, it's important that the girls get out into the yard to get some good ol' fashioned vitamin d (sunshine). We had been letting them out in spurts but decided to get more regular with our free ranging schedule. So now we release them for chicken happy hour in the yard after they are finished with most of their egg laying in the morning and they stay out until they come in to roost in the evening. This gives them hours of sunshine and foraging time but ensures that almost all the eggs are laid in the nesting boxes.
So these two things combined seem to be helping with the eggs. We haven't gotten anymore balloon eggs in the last few days. The Rhode Island reds are really out performing the ameraucanas. Here you can see all the eggs in my basket were brown.