No, they cannot, at least not political speech...the "sit-ins" at restaurants across the South decided that issue back in the 1960's. That was political "speech" on private property.
Yes, it was. However, prior to the Civil Rights Act, it was legal for those businesses to throw those people out at that time. It was legal for bus lines to segregate, it was legal to have separate drinking fountains and entrances to buildings and to segregate by skin color. People were forcibly removed, and, as attitudes towards segregation changed, those businesses suffered public backlashes, but they were still within their rights at the time to remove those folks from their property.
It was after the Civil Rights Act that it became illegal.
The problem is, Civil Rights are granted by government. They can also be revoked by that same government.
To add to my prior response, in most states with Concealed Carry Permits, you can be charged with trespassing (or worse) for carrying your firearm onto their premises if the business says that your firearms are not allowed or wanted on their premises. That would also be considered political speech, would it not?