LDS students may be endorsed only by the bishop of the ward (1) in which they live and (2) that holds their current Church membership record.
Translation: If you're a member of the church, go see your bishop.
Non-LDS students are to be endorsed by (1) the local ecclesiastical leader if the student is an active member of the congregation, (2) the bishop of the LDS ward in which they currently reside, or (3) the nondenominational BYU chaplain.
Translation: If you're not a member, go find any one of those three people to get an endorsement.
That person will hold a short interview and basically ask if you will live by the school's honor code. Say yes, and you have your endorsement.
Here's the specific form: Ecclesiastical Endorsement. Please feel free to ignore the parts specific to LDS applicants.
As I said, there's no requirement to be religious (only a statement "encouraging" non-LDS students to attend their respective religious services). Think of it like marriage: you don't have to go do a church to get married (although that's what religious people do), just go to the county building and sign a form or two.
But I don't think your problem is with the application process or with religion in itself: it's with the rules and regulations that go with being a BYU student.
You know what? That's fine: not even all "Mormons" want (or have to) live by those rules: I even know some that have a (gasp!) beard! They just go elsewhere for school, and nobody thinks the lesser of them.