I only know about Japanese Quail.
The eggs are sort of cream with pretty, brownish, splodge type markings. The inside of the shells is a blue colour. They do vary quite a lot. Some appear a bit bluish or greenish and we do have the odd white or green egg with no markings.
Your squishy egg was probably them just getting into the swing of things.
My dog adores quail eggs, much more so than hen eggs. He waits hopefully while I'm collecting them and they are the only food item that he will steal if I leave them on top of the pen for a while.