Yes, you can eat those eggs even when they're coughing and sneezing and have snotty noses. I just wouldn't recommend eating them while you are treating with antibiotics...IF you treat them with antibiotics. As you might imagine, while a hen is sick she'll generally lay less often and may even stop when she's feeling her worst. Once they recover, they generally start laying again although, again, they may lay less often.
Several posts back you asked about the possibility of food allergy. I think that's unlikely.
When I get a hen who's off her feed, I can usually get her to eat by making a warm mush of her layer feed and a little yogurt. For one hen I'd mix 1/3 cup of layer crumbles with hot water to make a thick mush and add 1 Tablespoon of yogurt. The sulfur in molasses is very good for a sick bird, too. I you have any on hand, you could mix in a teaspoon of that, as well.