Do you give them grit? Perhaps the sick hen has no grit in her crop to process (grind down) the feed.
By the way you know you're "training" her to follow you around by giving her bits of feed everywhere. I know this from personal experience -- just ask my spoiled rotten baby roo "Brownie".
As for the eggs, you fill a large bowl of water, then put the eggs in it. If they float, they are old. If they lay horizontally at the bottom of the bowl, they are very fresh. If they stay at the bottom, but one end of the egg rises (so the egg is vertical) they are still good, but not as fresh as the horizontal ones. This works because as an egg is laid with a small air bubble. As the egg ages, the pores of the shell allow the moisture inside to evaporate very slowly, therefore causing the air bubble to get bigger. The bigger the air bubble, the more the egg floats.
Hope this helps!