Some phrases on egg cartons give information about how the eggs were produced, but others are just marketing tricks.
For example, “All-Natural” is just marketing – the phrase does not tell you anything about the production process of the egg or even the egg. It basically states the evidence: that the egg is, in fact, a real egg from a real hen.
“Fresh Farm” and “No Hormones” also state the evidence. No hen that lays eggs in the United States receives hormones or hormone injections, which means phrases like “Hormone Free” or “Hormone Free” could technically be put on every carton in the store.