All chickens can produce great tasting eggs. The most important factors are what you feed them, proper care, and freshness.
As long as they get a balanced diet, which can be provided through layer feed (once they are old enough to switch from chick feed). When chickens are allowed to eat grass, greens, bugs, etc, their egg quality goes up, so if possible, free-ranging is beneficial, or you can provide these things inside their pen.
Egg quality goes down if hens are stressed or sick, so keeping them happy and healthy is very important.
Eggs will keep for about 1 month in the refrigerator, however, the fresher they are when you eat them, the better they taste. Those nasty things in the stores are usually pretty old when you get them, not to mention the hens never see a fresh piece of greenery or anything else fresh. Fresh eggs taste far better than those store-bought things.
Basically, just choose the hens you really want, keep them healthy, and you'll get great tasting eggs!!!
Lori