First of all, I don't believe that Amber knows anyone on a low-carb diet, everyone in her orbit is hugely overweight and not exactly a picture of health.
Second of all, people on low-carb/high-protein diets absolutely do eat chicken, it's one of the leanest sources of protein. Unless we're talking about people on ultra high-protein diets like bodybuilders, it's absurd to think someone on that type of diet will eat only pork or beef, but not chicken.
It's getting annoying how frequently she trots out the fact that she "never eats pork or beef, only chicken and turkey and lean meats" as if it makes her an expert on nutrition and healthy eating.