The right's being violated are both property rights. The taxpayer has the right to decide whether or not he wants his money to go to the healthcare of others. Government-run healthcare violates this right. The Catholic Church has the right to decide where they want their money to go, and forcing them to provide contraception is a violation of that right.

The optimal solution is that the government steps out and let's people retain their property.

I would argue that rights are never pitted against each other, though to make an argument for that I'd first have to see an argument to the contrary.

* Abortion.  You have the rights of a woman to her body and to decide whether to be a parent, the rights of a man to be a father if he got her pregnant, and also you can factor in the said rights of the unborn.  Of course, you can end up denying rights to one of the parties here and think you work around, but you end up missing nuances by only thinking in terms of rights.

* War. Innocent civilians do die, individuals who had nothing directly to do with a possible conflict, but who are part of a nation and contribute to the events in the war by their own personal actions by being part of the nation.  Like, the individuals who voted for the Nazi party and Hitler coming into power.

* The teaching of evolution in school vs parental rights.  Parents are to have rights over their children, but children are to have rights to correct information.

* Individual's rights to access goods and services in markets vs the rights of shopowners to serve who they want.  There are cases of racists collective belief systems that had African-Americans denied access to stores to be able to buy items they needed, or operate under the same set of rules as everyone else.

* An entire issue of community rights to social conventions people operate under vs the rights of individuals to do as they please.  Included in this is public property rights and access.  Take the example of Occupy.  They decide they want to use a park for camping as a civil rights protest, to make a point about Wall Street corruption and income inequality as part of first amendment rights.  You then face the fact the parks were not designed for such use, and the community opposed them being in there. 

* Smoker rights vs the rights of non-smokers.  Smokers argue they have a right to do what they want with their body.  Non-smokers argue they have a right to clean air.

* Property values of home owners being reduced by what their neighbors do.  In the case of a person buying property somewhere, and say the property next to it becomes a place that sells drugs or engages in prostitution, can result in one person's rights being violated at the risk of what someone else does.

Pretty much anything that isn't simply resolved comes about when you have rights in conflict.  When everything is merely framed in context of rights, there is a lack of yielding or seeing a larger collective good picture.  In short, you can't build an optimal ethical structure on rights alone or even one based mainly on rights.

Abortion: No one has the right to be a parasite on the body of another person, so that logically concludes that abortion is not a violation of rights. A woman has the right to decide whether or not she wants a parasite living off of her.

War: War doesn't validate the taking of someone's life. As far as morality is concerned, it's murder (though I suppose this is debatable).

Evolution: No one has a right to correct information. You do, however, have the right to question information that you are given.

Shopkeeper's rights: A store owner can refuse to serve whoever he wants for whatever reason. No one has a right to a good or service.

Community Rights: Communities don't have rights, they're abstract entities. If the Occupy movement has permission of the owner of the park (presumably the government, though I'd argue that their ownership is illegitemite) then they can protest there. If not, then they can't.

Smokers: Smokers have the right to smoke where the owner of the land/building says they can smoke. Owner's land, owner's rules.

It's not about whether or not rights come into conflict, it's about accurately identifying what rights are and which rights you have, something that has become increasingly muddled as almost no one outside of philosophers actually think about this anymore.

It's actually quite easy to build an ethical system based entirely on rights. Only one type of rights actually. Property rights, which are fundamentally the only type of rights that exist.

There are relatively few cases where rights are genuinely pitted against one another, unless you get into the highly problematic business of handing out positive rights. Most of the cases you mentioned aren't conflicts of rights, but of desires. They're mostly clear cut. So long as parents are legally responsible for their children, they can decide what their children are exposed to. When OWS is monopolizing a privately owned park that is opened to the general public, they can be forcibly removed. When a place of business decides to cater to or not cater to a certain group of people, that's entirely between them and their customers - laws compelling or forbidding them to do otherwise be damned.

Some cases are much thornier, though. The abortion example, for instance. As it stands, the woman is considered to have the full rights to terminate the fetus. The father has no say in the matter. But the second she squirts the thing out, the father is suddenly on the hook for child support. That doesn't really seem fair, but then again, neither do most of the alternatives.

The case in point is certainly not one of those cases, though. A woman has a right to obtain contraception from any willing provider. The Catholic church is not willing, yet they are being coerced into doing so anyway. So it's pretty clear whose rights are being violated.

Just wanted to share the opposing view... If you argue that the fetus is a life, then you would be placing the rights of the woman over the rights of the baby. In the case of normal sex, the woman's choice was made when she chose to have sex with her partner, and thus has no right to have an abortion as an abortion would be an act of aggression against the fetus. In the case of rape, the aggressor is the rapist... But you can't punish the fetus for a crime that it didn't commit, as that would again be an act of aggression against the fetus. 

The problem in this case is that the child can't seek damages for such an act of aggression (again, this is assuming that you view the fetus as a life), and so prosecution would depend on people who want to act on behalf of the unborn child. So abortion does violate the non-aggression principle for someone who is pro life, but actually enforcing that is a tricky situation.

And another weakness involving rights-based ethics systems is the issues with defining what are rights and what aren't. It is like, so long as someone can argue for something, they can make anything a right. Pretty much anything can be argued a right. The abortion issue rests in this area of having this problem.

And then by framing everything in terms of rights, issues which do end up resonating as having ethical concerns, and need for restrictions, get framed as rights. People think being cruel to animals is wrong, and that destruction of the earth and pollution, are both issues that need to have the ability to call ethically restrictions. When you have a rights-based ethical system, you end up giving rights to animals and the earth. One might go animals, but then the planet earth has rights? What is a better framework is to call for duties and responsibilities, along with said rights. But, by being rights-based, you have no framework for doing that.

A rights based ethical system handles this quite easily ...

An individual has the right to engage in sexual acts with other consenting adults, and they have the right to use contraceptives if they choose to, but they have no authority to force someone else to pay for their contraceptives; or to have other people pay for the children they have or the diseases they get.

Rights based ethical systems are based on the balance between individual rights and personal responsibility. Unfortunately, those who are constantly trying to eliminate individual rights do so because they fail to accept personal responsibility for anything.

Again, in a rights based ethics system you have the right to be left the fuck alone and you can't possibly have the right to anything that someone else has to provide for you. What you're describing is people raising their desires to the level of a "right", and that's not a problem of ethics but of politics.

If the entire basis of human interaction is to be left alone, you have little in the way of human action.  There is no such thing as love in this system either.  And there is nothing in this that manages to help people who want to change.  There is no call to do good or serve others.  It is entirely self-focused.  There are no Good Samaritans either. And that which people consider to be noble is gone.  In the world you have, Superman would go in to business for himself, do celebrity route, become famous, and people would end up dying.  

I didn't say that it's the entire basis of human interaction, just that it's the right you have. Every negative right boils down to just that. So you have the right to speak freely, and that right shall not be infringed upon no matter how unpopular your ideas are. However, you do not and cannot have the right to a forum from which to express your ideas if it means that someone else has to provide that forum for you.

Likewise, just because some freeloader decides that free birth control is a "right" doesn't make it a right.

Where is the room for good Samaritanship in a system in which all charity (and the responsibility to do charitable works for those who believe in such a thing, like Christians) has been made the domain of the government?