First we need to agree on what a right is. Then, where does it come from? Then, who enjoys rights.
I would say a right is a condition of existence required by mans' nature to assure his survival. The three fundamental rights are life, liberty and property. I would argue that rights are endemic to the individual, not to groups. I would argue that when the rights of the individual are protected, then the rights of everyone are protected. I do not see how rights can come into conflict with each other.
Let's look at property rights, the right/ability to privately own property. I assert that ownership of property is defined by who gets to determine what is to be done with said property. If the government imposes its will on an owner such that the owner has to dispose of the property as the government demands, then in reality it is the government that owns the property. When the government forces bigots to share their property with those whom the bigots despise, then the bigot doesn't own the property, the government does. The bigot's right to property has been denied. One of the three fundamental rights has been violated by government decree. Now, apparently you think that is morally justified, that bigots should have their rights violated. They're hateful bigots, they have it coming. I don't disagree that bigots are hateful. But I argue they have the same rights as non-bigots. When you justify denying rights to one individual or group you climb onto a slippery slope from which it is almost impossible to escape. The Civil Rights Act of 1964, at least those two sections, violated property rights, pure and simple. Something was gained, but something else was lost.