Judge says Education Dept partisan out-of-office emails violated First Amendment

I fundamentally don't understand the rights of government employees. They are supposedly there to execute the will of the political branch that controls them whether or not they agree with it, which is why they are given immunity from firing by each incoming administration. So how do they also have the right to personalized communication from their work email addresses (a right that no private sector employee has)? How can they have the right to exercise government authority without being democratically elected, or at least accountable for their actions to someone who is?

Can you explain what you mean, in more direct or simpler terms?

So the judge's quote from the article is "and they certainly do not sign up to be a billboard for any given administration's partisan views."

I thought that's exactly what you signed up for when you become a government employee.