Not sure what eggs to buy or what free-run, cage-free means? Hopefully this will help:
Brown eggs: Eggshell color has nothing to do with the quality, flavor, nutritive value, cooking characteristic, or shell thickness of an egg. Shell color only depends on the breed of hen.
Omega 3 enhanced eggs: Eggs are from hens fed a diet flax seed or fish oils. Some experts say the omega 3 is unstable and quickly becomes rancid.
Organic eggs: Eggs produced by hens fed certified organic grains without conventional pesticides, fertilizers, growth hormones, or antibiotics.
Free-run or Cage-free eggs: Eggs produced by hens that are able to move about the floor of the barn and have nesting boxes and perches.
Free-range eggs: Eggs produced in a similar environment as cage-free eggs but hens have access to outdoor runs as well.
There is a lot of controversy about nutritional value of conventional vs. organic eggs. Controversy also surrounds the color of the egg yolk. The color of the egg yolk depends mostly on the diet of the hen. Hens that eat grains and cornmeal tend to have lighter-colored yolks. Hens that eat plants, produce, and grub tend to have darker yellow to orange-colored yolks. Eggs with darker yolks are reported to be healthier and tastier. Some studies have reported chemical differences and others have not. Even if free range eggs are not healthier, I believe that the living and food conditions offer an advantage. I love seeing the different hues of the shells and size of the eggs.