This isn’t first amendment issue. Until the government of certain cities shuts down Chic Fil-A, the first amendment isn’t involved. To reiterate the 1st amendment: “CONGRESS shall make no law…abdriging the freedom of speech.” The Fourteenth Amendment applied this to the states. So, basically all state, local and federal institutions must respect this right. But, this doesn’t necessarily apply to private actors unless such an institution has some public aspect to it (like people can protest in private shopping malls) or gets public funding.