You can refuse to sell your property to a variety of people--anyone with tattoos, or instance, or people who wear glasses, or people who own canaries, or people who wear hats indoors.
However, there's the whole sticky issue of Fair Housing. And it's particularly sticky because the test is not whether you (or your agent) were discriminating against someone, but whether the potential buyer believes that they were being discriminated against.
For instance, to use the examples I gave above: Suppose the person with tattoos is a member of a protected class and believes you were discriminating against him on that basis? Or people who wear glasses: Maybe the person's vision is 20/50 and they wear reading glasses. Probably not a problem. But what if the person's vision is 20/300, and they're legally blind? Now that might be a problem. Discriminating against a canary owner: Probably OK. Pet owners in general: OK. But what if the animal is a seeing-eye dog? Uh oh; that's a real problem. Or people who wear hats indoors? Maybe you don't like "uncouth" guys who wear baseball caps with the brims in back. That's OK. But what about the various religions that believe in covering the head?
Again, all it takes to initiate--and to pursue, sometimes successfully--an action is a belief by a person that he or she has been the subject of discrimination. So, you must be very, very careful.
Discuss your concerns with your agent. He/she probably will want to discuss your specific situation with his/her broker in order to come up with a solution that addresses your concerns yet steers clear of any possible hint of discrimination.
Hope that helps.
You certainly can set the rules and expectations for showing your property. There are situations where a lisitng agent must even see a written pre-approval before showing the property.
But, you need to weigh the pros and cons of any decision that you make. There will be some buyers, and some buyer agents who may decide not to see or show your property, because they wish to not divulge that information. Their privacy concerns may have nothing to do w/ you or your property. A buyer might simply not want it known that he or she is looking. Perhaps they are looking and have not told their children and fear theif kids could hear about their visit. There could be any number of reasons that would bring a buyer to decline. Most buyers will comply, but what happens when it is the buyer who really might have written an offer?
It is your house, and your right to ask for this info. Simply weigh the value to you and make sure it outweighs any potential lost showings. You can determine to exclude a specific buyer, as long as it does not violate a protected class as defined in the local, state, or federal laws governing your area.