The law doesn't distinguish between two Christians with divergent beliefs, or between an atheist and a Christian (obviously with divergent beliefs). The law simply does not care what religion you have, or whether you have one. The law just says "follow the law!".
The complication is that part of the First Amendment which says that the law is to be neutral as to religion also has what's known as the "Free Exercise Clause", the part that says "or prohibiting the free exercise thereof", which has been taken to refer to actions undertaken because of that belief. There have been various rulings on the conflict between religious doctrine and laws requiring or prohibiting certain actions (mandatory flag salutes, conscription, religious pamphleting). When a law conflicts with a fundamental right such as a right protected by the First Amendment, such a law is allowed only in narrow circumstances (known as strict scrutiny).
It is up to Congress to state what kinds of First Amendment-based exceptions there are to laws. In a case involving the draft, SCOTUS held in US v. Seeger, 380 U.S. 163 that
The test of religious belief within the meaning of the exemption in §
6(j) is whether it is a sincere and meaningful belief occupying in the
life of its possessor a place parallel to that filled by the God of
those admittedly qualified for the exemption
This view underlies the current regulation on employment and religious discrimination in 29 CFR 1605.1 that
In most cases whether or not a practice or belief is religious is not
at issue. However, in those cases in which the issue does exist, the
Commission will define religious practices to include moral or ethical
beliefs as to what is right and wrong which are sincerely held with
the strength of traditional religious views.
The question that EEOC or the courts would have to answer is, what fundamental life belief is being violated by compelling a certain action? Forcing Muslims to eat pork violates a fundamental belief of Muslims. Prohibiting Muslim employers from discriminating against pork-eaters does not violate those beliefs, because there is no fundamental life belief held by Muslims that it is a mortal sin to hire pork eaters.
In this case, Ann is at a disadvantage because she can't maintain that being forced to hire Christine violates a fundamental belief of hers (it's like refusing to hire pork-eaters). If Christine was obnoxious in espousing her viewpoint in a manner that reflected badly on Ann's business, Ann can fire Christine. Ann might, on those same grounds, refuse to hire Christine if there was a well-justified belief that such damage to her business will result (you don't have to wait until your business is destroyed). The (implied) fact on Christine's resume is not sufficient evidence that Christine will cause a problem for Ann's business. Instead, it is a plain instance of religious discrimination, which is prohibited by law.
