Are there any cirstances where someone can be fired from their job due to there beliefs?

I have a few hypothetical questions here... Let's say a gay person got a job at a conservative non profit organization that was vehemently against gay marriage and against any gay rights... If this person's employer found out they were gay could they be fired as it looks like a pretty clear cut case of conflict of interest. Or the other side of this... what if a person that was very conservative and believed gay rights were complete BS got a job at an organization that promoted gay rights or gay marriage. If it was found out that they held these beliefs could they be fired legally. Finally, what if someone was an outspoken Nazi and 4 months after being hired by ANY company or business (and at the time of hire the company had no idea about the employees Nazi beliefs) their boss sees them giving a pro Nazi speech on youtube or something to that nature. Could that person be legally fired for bringing bad publicity to that company even thought they are simply exercising their first amendment right outside of their work environment. (I know these situations may never arise but im just curious). Thanks.