Yes, Navajo eat meat. There is no problem at all with eating meat though they would probably avoid eating certain types of meat like snake and etc. A couple times a year, the family goes out to butcher a lamb (or two) to provide extra meat for everyone. For instance, although my father-in-law may not live on the ranch, he still will get a portion of the meat in the form of a leg or other piece. Life and death are part of nature so killing a creature for its meat is of no offense. It is the way things are. However, it is expected to thank the fallen animal for their meat and so on.
The Navajo try to "walk in beauty". It's a very hard concept to explain. A loose way of explaining it is that we are all a part of nature and we should live in it with appreciation of acknowledging that which gives us life (and not just the mother but EVERYTHING) and living in harmony with it. For foodstuffs, acknowledging that all things have a spirit or a deity attached to them does not mean that it is forbidden to kill or even harvest things for food. Doing so would be unnatural. It is natural for the Navajo to need meat and other foods. It's just knowing that these things give them life and respecting them for it. There are some things that are avoided being killed or parts of certain animals--if killed for food--aren't used as well but those are considered things that are associated with evil or would anger a deity.
It's a really hard topic to explain. I hope that some of what I wrote made sense. It's an idea that I understand but putting it down in words is hard, lol.
Stephanie
In every man there is something wherein I may learn of him, and in that I am his pupil.--Ralph Waldo Emerson