Myth:
When atheists try to prevent government officials from promoting, endorsing, or encouraging their religion, they are violating the officials' constitutional rights. Government officials have a right to free speech and a right to freely exercise their religion.
Response:
The right of individual citizens to promote their religion and proselytize others is protected in two ways: first as a form of free speech, and second as the free exercise of one's religion. Everyone is protected equally by these rights, so doesn't this mean that government officials have a right to promote and endorse their religion? Many on the Christian Right think so, but their arguments fail because they completely misunderstand the nature of individuals rights vs. government action.
What may be most important to understand is that a government official while exercising their official duties is not acting as a private citizen anymore. They are instead acting on behalf of the state. When acting as an officer of the state, a person simply doesn't have the same freedom of expression that they do as a private citizen. A private citizen can say pretty much anything they want; an officer of the state must instead represent the interests and views of the government itself.
When acting as an officer of the state, a person cannot say "Islam sucks, Christianity doesn't." They cannot encourage people to convert to their religion, to read their religious scriptures, or to attend any particular religious congregations. Hardly anyone disputes any of this, but obviously a person can say such things as a private citizen. So there is clearly a difference, agreed upon by nearly everyone, between what a person can do privately and what they can do as a state official.
This distinction doesn't only exist for extreme cases like those listed above. For example, a person like Roy Moore can put a Ten Commandments rock out on his front lawn, on the side of his house, or in front of his church; he cannot, however, do the same for government buildings. The reason why is that the government does not have the authority to promote or endorse the Ten Commandments in any way. Private citizens, however, can do that all they want. Roy Moore's rights as a private citizen were never infringed upon.
So a person doesn't have a protected free exercise right to promote their religion while performing their official duties, but what about free speech? Don't government officials have broader free speech rights to say what they want about what they believe? No, they don't, and the example of police officers may be the easiest way to explain why. Police officers dont have a free speech right to say absolutely anything they want in public while performing their official duties. A police officer who pulls you over for a traffic violation can't comment on your clothing choices, your ethnicity, or any other personal characteristic which is irrelevant to the matter at hand.
If a police officer says the wrong thing while performing their duties, they can be disciplined and even fired. Is this unconstitutional? Absolutely not. The same example can be used with religion: a police officer who tries to witness to you for Christ while he gives you a speeding ticket is not exercising his freedom of expression, nor is he exercising any right to religious liberty. Preventing him from evangelizing and proselytizing while on the job would not violate his First Amendment rights on any level.
At the same time, he can do all the evangelizing he wants while wearing civilian clothing and standing on a street corner on his own time. That's a parallel to what we have when a government body opens an official meeting with an official prayer to Jesus Christ, or when a judge installs a Ten Commandments display in a courtroom. When government officials are acting in their official capacities, they have to stick with the government's public business and set aside their personal religious affairs. That's part of the price of serving the public in the government.

