Is there a difference between fresh eggs and store-bought eggs? It depends on what you feed your chickens, how healthy they are, and how you keep them. For example, birds that are full of parasites will have WORMY eggs (mine don't--that would gross me out!).
We add flax seed to their diet--they also get the shells from the eggs (calcium) and free-range the chickens in addition to giving them grain. They love stale bread and yogurt/sour milk. The yolks are a richer color. The whites are not rubbery (I don't have any store-bought eggs with with to do a cooking test, but it seems to me it takes less time to fry a fresh egg than a store-bought one and the whites are not rubbery). Fresh eggs don't peel as easily--there are tricks to peeling fresh eggs. When you hard boil a fresh egg, the air pocket is much smaller than most store-bought eggs. My chickens are happy chickens--they get outside, they get to forage, they are not kept in small boxes...they also were scared today when a hawk showed up. I was scared too, but when the "sentry" hen called out the alarm, the rooster and other "top" hen joined in and chased everyone into the barn and spots of cover before they also went in the barn. The new flock had no problem lining up and getting into the coop before the hawk could figure out if it could swoop down between the trees to get to the hens--we put the coop under cover of some trees for shade in the summer--but it also gives them some protection because they are not out in the open. There must be some protective gene in them--they know enough to seek cover when a predator shows up. Then a stray cat showed up...the poor hens had a stressful day.