This is not about religion o the government. It is about why healthcare should pay for birth control.
Health insurance is not a magic pool of money - any money you get out must first come in. If birth control costs say $30/month, then if everyone buys it through insurance, it raises the cost of insurance by say $32 - after all their are administration costs and and the insurance company needs a profit.
So, why do it through health insurance as opposed to paying for it ourselves? It clearly costs more
Well, first of all, not everyone buys birth control but everyone benefits from it. When we do it through insurance, it doesn't raise the women's health care cost by $32, it raises both men and women's healthcare by $16. This is fair because the men benefit as much as the women do. For every woman that has an unwanted child, their is a man with an unwanted child as well.
But that's not all. If the birth control is condoms, we not only control birth but we limit the spread of deadly, incurable, EXPENSIVE diseases. Condoms are clearly healthcare - and as anyone concerned about an epidemic would consider it worthwhile for all to use birth control. Birth control pills have similar, though not as strong or obvious, health effects.
Also, if it is covered by insurance than that encourages people to actually use it - as they have already paid for it. It stops idiots from having unprotected sex because of short term money issues. Anything that stops idiots from having unprotected sex is a good idea. We have enough idiots, we don't need them making more.
Moreover, as China has clearly demonstrated, it is in the interest of United States to limit birth. While China is a autocratic country that uses evil methods to limit births, that does not mean a democratic country can't use legal insurance rules to encourage people to limit births.
That is, it is in the states' interest to reduce birth in the USA . Our population is growing and excessive population growth can cause problems (just as Japan is having an issue with insufficient population growth)
As such, it is reasonable to charge everyone to control births, even if they don't want to do it. Requiring insurance companies to offer it is a kind of indirect tax on consumers that don't want birth control. It makes them pay for those that do. Which is currently in the United States's national interest.
Note, if we ever have Japan's problem of insufficient births, it would then make sense to prevent insurance companies from covering birth control, and perhaps to even tax it.