I'm not sure it's just in the Midlands. My wife came from Darlington, and in her family hens were always 'chooky hens' (to rhyme with 'hooky') and eggs 'chooky eggs'. I assumed it was just childspeak which had been continued (as all rabbts are 'bunny rabbits'), since there were always young children from somewhere in the family liable to call round at any time.
Nor do I. I remember a computer game called 'Chuckie Egg' which is obviously about chucking eggs. I have the feeling that soft eggs are called such because they more likely to break than hard-boiled.