By this logic, state employees and elected officials, who are themselves US citizens, have the right to say anything at all to the public, including blatant lies about public health issues, because there's no specific legal exception banning them from doing so.
They are only merely citizens outside of work, and they are indeed that free but only during those times.
When they are at work acting as the state, they are the state and their actions are limited to the rights of the state.
This should not have been something that needed explaining.
As government representatives, they’re limited to the behavior that the Constitution grants to the government. Here’s an explainer to said Constitution you may find useful: https://www.law.cornell.edu/constitution-conan/article-1/sec...
After reading that, please do tell where the power to poison water and lie about it is granted to the government, and the members of that government.