It's a very uncommon and awkward way of saying it. The only distinction I could determine is that by deviating from the typical "I never eat meat", you are placing emphasis on the differences in your sentence. By saying you "don't ever eat meat", you are not only conveying that you never eat meat, but that you don't ever, placing emphasis on the "ever" part by isolating it from the "not" (which is implied in the "n-" prefix at "never"). This may connote revulsion or insistence that one does not ever eat meat. Usually in speech, there would probably be verbal emphasis on "ever", hence the awkward wording. For example.
Person 1: "Do you have any unique parts about your diet?"
Person 2: "For one, I never eat meat."
Person 1: "Have you tried that steak over at the local grill? It's delicious!"
Person 2: "I don't ever eat meat, so I wouldn't know. I've told you this before."
The point of this lesson is to illustrate how to use adverbs. The adverb in this case is "never", so it is essential to use it in the sentence. There is also a slight difference between your sentence and the correct sentence in this particular exercise: yours states that one does not eat meat, but does not specify the duration of one's abstinence from meat (although it's typically implied); the sentence above specifies that one never does, which has a stronger connotation than simply not eating meat.
It depends. Very often the order is indifferent. But when a negative adverb follows the verb, it requires a double negative by prepending no to the verb:
“Yo no como nunca carne”.
Note that this double negative does not result in a positive, it still means "I never eat meat".