Why are certain beliefs privileged?
Because the people who founded this country came here seeking relief from religious oppression. Thus, when they created their own government (the one we have today), they ensured that the highest law of the land specifically restrained the government from doing to future generations what the Crown had done to them. If you don't think religious beliefs deserve special consideration, feel free to propose an amendment to the US Constitution stating so.
Could a non-religious person decide they "believed" in not providing certain healthcare to their employees and just let the government pick up the bill instead?
That would be a more challenging case to prove. The benefit of belonging to a popular religious group is that the tenants are widely known. As such, one must only then demonstrate that one actually belongs to that group (and even so, only minimally; stating as much without evidence to the contrary would typically be enough) to gain protection from government policy, law, or action which would violate that group's religious beliefs. In the Hobby Lobby case, there were 4 specific methods of birth control out of 20 which the owners maintained violated their core beliefs. In essence, they viewed those 4 specific methods as murder, but raised no objection to the other 16. The SCOTUS found those beliefs to be sincere and reasonable, and found that there was no interest at stake compelling enough to override the protections afforded to the owners of Hobby Lobby by the US Constitution. This was found in no small part due to the multitude of other options available for those seeking to attain the goals of the underlying legislation.
It's actually a pretty mundane case and shouldn't get people this riled up, but it does because the ACA and the President are attached to it. If this case involved any other law but the President's signature legislation, nobody but SCOTUS buffs would have heard a word about it.