I find that the organic eggs are on the verge of inedible/B-grade. They are too runny the very day I purchase them often, let alone after a few days in my fridge. I buy organic whenever I can but I find that my experience here with the ones from Whole Foods/other grocery stores is always poor. I always buy the Grade-A eggs (I can never find AA organic). Can anyone else suggest good brands?
The proteins in the egg white and around the yolk degrade over time. They degrade much faster at room temperatures than in the fridge. I have read that a day on the counter is equivalent to a week in the fridge. This matches my personal experience.
When the proteins degrade the white will be runnier and yolk will be fragile and not be as taught. Old eggs are better for boiling exactly for this reason. The shells come off more easily.
The diet of the chicken will play significant roles in the strength of the shell and the nutrition of the yolk but none at all in whether the egg is runny.
Similarly- the fact that they are "organic" (whatever that means) will not effect the runniness.
If the eggs you are finding are consistently runny then it is because they have either been sitting on the shelf too long or they spent too long at room temps. Either find a store that turns over its stock faster or a brand that handles the eggs better.
I'm going to go simpler...checking dates on your eggs goes a long way. The other answers have been totally correct, but I've found that even a high volume store like Walmart will sell eggs that are nearly expired.
I have a couple different brands of eggs that I buy and pick strictly based on the one with the furthest out expiration date. Look through the eggs and find one that is significantly far out. I've seen a cartoon of eggs that expires in 3 days sitting next to a carton of eggs labeled to expire in 3 weeks. It's craziness!
Checking the dates will get you a good chunk of the way there.