Should a business be allowed to not let people into their store/restaurant if they are carrying a weapon?

In the case of Texas, it's completely allowed. All they have to do is post a 30.06 on the door:
Also you can't carry in a business that makes 51% of it's profit selling alcohol.

No signs, no complaining. If there's a sign, only police are allowed to carry.

It should be up to the business owner. It is their business and if someone doesn't like it they can spend their money elsewhere. I would think it would fall under the "We reserve the right to refuse service to anyone." statement.

Healthcare facilities do not permit non law enforcement personnel from having a weapon in their possession in their facilities

Sgt Kelli Mays the fact that Texas has passed an open carry law should not be construed to mean it is a mandate for every citizen who has a gun to be able to go where and when they please while carrying a weapon. If open carry was mandated it would be trampling on the rights and responsibilities of other citizens, shop owners, schools, houses of worship and groups of people who either don't feel that open carry is appropriate where they are as well as those who have been traumatized by gunfire.
I support open carry and closed carry where it is legal; but, I don't consider that a mandate to trample on the rights of others. We each need to be wise how we exercise our "rights" to ensure that we are responsible.

Absolutely a private business should be able to prohibit firearms on premises. You dont have to patronize them. I am a concealed weapon permit holder in South Carolina. Here proper signage must be in place.
That is the great thing about this country!