As far as I know, anybody from anywhere can buy anything they want in America as long as they can afford it (barring weapons, radioactive materials, and other controlled substances). As long as you give your proper address so the tax bills can be mailed to you, you should be able to buy and keep any property.
As far as living on this property, you should consult this US government website about visas and immigration.
http://www.usa.gov/visitors/visit.shtml