I have a 25 week old delaware that sometimes lays those "squishy' eggs. I had never considered eating them - they are just too weird. What I have noticed is that if she lays one of these, it is late in the evening - and, she doesn't lay the next morning. The eggs she does lay in the morning have shells that are nice & hard. In my mind - I think that these squishy evening eggs have passed through her too quickly, and the shell hasn't had time to properly form. If she doesn't drop one in the evening - by the next morning the shell has had a chance to form properly. The evening eggs she just drops wherever she is, hard shelled morning eggs are always in the nestbox. I will admit I don't fully understand the entire egg forming process, so, this may not be correct, but, it seem logical to me.
She has been laying for about 5 weeks now, I am hoping this problem will resolve itself - maybe she needs to do her kegel exercises.....