I used to work for a major pharmaceutical company. Not only did they perform animal testing, but they did it in the facility that I was working at. The person who interviewed me had the courtesy to inform me of this up front during the interview, so that if my beleifs conflicted with animal testing, I could take it under consideration if offered a job.
However, as your example make clear, there are some deal breakers that the interviewer won't tell you about, either because they are not thinking about it, or because they din't realize that it could be a deal breaker.
That being said, you still do not need to step outside of the EEO boundaries. The way to approach this is to either simply mention "personal beleifs", or to ask directly (or indirectly) about the deal breakers without saying why. The problem is that 95% of people feel some inane urge to explain the irrelevant. People have this unexplainable desire to explain in great depth what can just be left at "personal beleifs". Like, "I am a strict Muslim and I will not work for a pork processing plant" is way too much detail. Being Muslim does not need to come up. Just say, "personal beleifs", and the interviewer still does not know your religion, and the point is just as clear.
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