Whether an employer can discipline or even fire an employee for postings on a personal social media account that convey a message inconsistent with the values of the organization may depend on states. Technically, employees working in the private sector have no First Amendment free speech rights because the Bill of Rights only applies to government action. But state laws can also grant employees free speech protections.
Levy Employment Law Blog:
Free Speech at Work: Employer’s Enforcement of Social Media Policies May Turn on State Law