I think that the red dye is likely the least of your concerns. If you find that unsettling, I would avoid investigating things further or be prepared to go vegetarian or organic free-range meats exclusive.
If you have ever been to a meat or chicken processing plant, it is not something you will easily forget. I have nothing against eating meat (as a matter of disclosure I am a vegetarian, but out of habit from my upbringing rather than deep conviction), but the shear amount of meat that we eat as a country encourages some pretty crazy stuff in the industrial scale farms.