Well, this isn't the first time, in the movie at least, Ennis expresses his concern for the job. When he rans into the bear and the mules ran away, scared, dropping all the food, Jack proposes to kill a sheep. And Ennis says "We're supposed to guard them not eat them". I don't think he was being ethical or anything, since he had no problem killing an elk on a preserved forest land. I think he was a bit afraid Aguirre might find out about it.
Just a note on cultural attitudes, in general: to a ranch kid, there's a huge, huge difference between killing a wild animal and killing a domesticated one. National Forest land isn't really "preserved" in the US -- it's officially run by the Department of Agriculture, and it's supposed to be the "Land of Many Uses." It's land that wasn't successfully homesteaded, for some reason or another. (And I still don't have a clue why the Bureau of Land Management ends up in charge of some land, and the National Forest Service runs others. And there's BLM land right beside my house, too.) Anyway, I think that urban and suburban Americans are beginning to see National Forest land as something that exists for environmental preservation and/or for recreation, but the longterm residents of the rural West (and the current government) see National Forests as land that exists to be used, whether for hunting or grazing or mining or extracting oil and gas.
Anyway, hunting on Forest Service land isn't a big deal. Odds are that Jack and Ennis have both gone hunting for something or another many times before. The only issues with hunting the elk are that 1) they didn't have a hunting license, and 2) they were shooting it out of season. (Elk season would be in the fall, right around now, around when the snow starts falling.) So legally they were poaching (thus the comment about Game & Fish catching them), but in terms of perceived violations of the local ethical code? It's not a big deal, not in the way that shooting a sheep would be.
Edit: And to get back on track, I agree with Katherine here -- that this is a case where the line is both very in character for Ennis, poor and wanting to be able to keep working, and also serves to make the audience forget that the relationship on the mountain is only a temporary thing. And the choice of the word "we" is important there.
I don't know if Ennis is actually thinking that the relationship will go on forever, but I think that Ang Lee and Diana Ossana and Larry McMurtry are deliberately trying to lull us into believing that it will.