The answer to this question is always Controversial. But I clarify each and every aspect for better understanding. Basically organic eggs are the eggs, which are produced by the birds, who are eating organic Food without any antibiotics /anticoccidials or any other hormones etc.
Second thing is organic eggs should be obtained from the birds reared in Open range not in cages. It is considered the bird in cage remain in stress and the quality of egg may be lesser
But nowadays, some groups are keeping the birds in cages and providing feed without any Antibiotic/anticoccidials and other .They are giving feed to birds with additional herbs. Which increases the taste and quality of the egg. But in this case the bird is not in free range.
Huge demand for organic eggs is fulfilled by these types of eggs. Outer Colour of the egg may be brown or white. It only depends on Breed of Bird.
Now the most important aspect of quality. The birds in cages/commercial Deep litter are given minerals and Multivitamins that also give impact on eggs. What the bird is eating, the result will also affect egg quality.
But in both cases either bird is free range or not, if the bird has not been fed Antibiotic/ anticoccidials and additional herbs given to birds along with poultry feed .Its quality will be better.
Real organic egg is when the egg got from the bird reared in the open and having a natural diet.