The US Constitution generally limits what the government can do/restrict.
No, it doesn't factor into it at all. The First Amendment reads as follows: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances." Bolding added. Note the bolded first phrase: "Congress shall make no law..." The First Amendment only protects against federal government intrusion on those listed rights. The Supreme Court has held that those rights now apply to protect against such intrusion by state and local governments, too, through the 14th Amendment. But the First and 14th Amendments do not extent the protection to private persons. As far as the Consitution is concerned a private business is free to restrict the rights of its employees with respect to free speech, religion, assembly, etc.
The only reason that employers cannot discriminate based on religion is not the Constitution but rather the Civil Rights Act of 1964. But there is no similar federal statute that protects employees from an employer restricting their speech or right to assemble. Few states have such laws either. However, there is at least one state that has a law that would protect the OP here: Colorado. Colorado law makes it illegal for an employer to discriminate against an employee for any out of work activity of the employee that is not illegal.