The original concept of natural rights was "life, liberty and the ownership of property", this morphed into "life, liberty and the pursuit of happiness" when it was "sampled" for use in the declaration of independence.
Property ownership is a fundamental right. Life and liberty are also fundamental rights, and both of these are significantly empowered by the right to keep and bear arms, to state the obvious.
So we have a clash of fundamental rights here. You have the right to bear arms. A business owner has the right to the enjoyment of their property. According to our Declaration of Independence, the whole point of having government is to support and defend these natural, or "inalienable" rights. But what to do when there is a clear conflict of rights?
So far, the government's answer has been that property owner's rights trump individuals rights, unless of course, we are talking about the right to have a cake made for your gay wedding, or the right to eat at the same counter as people of a different race. The government views these as more important than the right to property ownership, and so property owners are forced, ultimately at the point of a gun, to give in to these demands. A citizen's right to keep and bear arms is viewed by the government as being less important than these other rights so property owners are not forced to accommodate this right. At least for now.
Personally, I think that racism and the irrational fear of guns both stem from ignorance. The solution for ignorance is education, but unfortunately that will not happen when you have a significant percentage of the population that is still ignorant. This was the pattern with racism, and hopefully we will see a similar pattern with the irrational fear of guns.
In addition to natural, or God-given, rights, we also have the concept of Liberty, which can both cloud and resolve issues where rights of different folks are at odds. Liberty requires of us that we generally do what we want to do only to the extent that our activities don't infringe unduly on the rights of others. When our actions begin to interfere with other people exercising their rights, we must consider the circumstances and whether our needs/desires can be met without burdening others, or if the burden we this impose on others is the least onerous that the situation allows.
In cases of businesses putting out the Un-Welcome mat, so to speak, they are asserting their property rights. If I consider the exercise of my right to defend myself with a firearm equally important to their right for me not to enter their establishment with one (or perhaps more important), I have a duty to myself to not be disarmed. Given this duty, though, I must still act within other valid and appropriate laws. I do not have a duty to set foot in a posted business or spend money there, and if the owners have made their intent plain via correct signage or other means it is unlawful for me to enter their establishment. I therefore simply move on and do business with others who do welcome my business.