I feel for you. I used to work at McDonald's. When I started, I wasn't actually vegetarian. However, by the time I finished, I was vegetarian and communist (not that the two are related). So for one, I was working for the "evil corporation" and for two, I was contributing to the slaughter of animals.
I worked quite awhile before quitting. I actually quit because of other reasons rather than some great moral act. Honestly, a person is really kidding themselves if they think they can be some sort of moral hero. All you can do is try your best.
Should you quit or find another job. I struggled with that myself. Ultimately, would being without the guilt of working outweigh the benefits you get from working. I'm not sure ethics are much more than feelings. That's if ethics are simply based on feelings. If they have a greater rationality to them, the question is even harder.
Really, I'm not a person with moral answers for you. I'm vegetarian, and I'm not sure eating meat is wrong. Killing animals is wrong. Technically, people are demanding a service, meat, and seen as somewhat responsible for the death of those animals. By that reasoning, providing the service might make you somewhat responsible. I don't know.
Does the allowance of people to make their own dietary choices outweigh other issues here. Is the only person truly at fault the one who is killing the animal. I have no answers to this issue, and I study philosophy, my primary interest being ethics.
Economically speaking, you don't play an essential role in the system, you leaving will have someone else fill your place. Demand is fixed and independent of you. Individuals wanting the profit will take your job or the price of meat will go up. If the price of meat goes up, in my opinion, conditions of animals would worsen. You "might" even be helping animals in the long-run.