Publix Company Policy: "Publix respectfully requests that only law enforcement officials openly carry firearms in our stores." September 5. 2019.
Publix has changed their policy recently. Not no open carry, but no carrying at all. September 15, 2015.
Publix Company Policy: Allows handguns, but, they do not allow long guns. Furthermore, Publix prefers concealed carry of handguns, but depending on the store, the store manager may tolerate open carry. July 3, 2008