The onus falls on the one making the first claim to explain how the government could possibly have no free speech rights in the first place.
But fine:
The government - all three branches of it (judicial, legislative, executive) - is not some nebulous entity made up of some nebulous actors. The government is made up of people, more precisely, citizens of the United States, a.k.a Americans, not just by feeling - but by law as well.
Americans, as you may - or may not - be aware, have free speech rights as defined in the Constitution of the United States - the First Amendment, if I'm not mistaken.
Americans, when fulfilling certain qualifications, can one day, become part of the government, whether that's going to be a more visibly important position (like, say, the POTUS, a Supreme Court Judge, or a Senator, or a Governor of a state, or a Director of some agency, or other positions that I did not mention), or a, relatively speaking, menial employee, working for some government agency.
The President has free speech rights, the judges (in all courts) have free speech rights, the Senators and all other law-making individuals at the federal, state, or any other level, Governors have free speech rights, "menial" employees, and all others, have free speech rights.
Their free speech comes from the fact that they're, one, American citizens, and, two, on US soil, thanks, in part not just to the First Amendment of the Constitution, but also to the long-practiced precedent of freedom of expression expressed on that particular part of the world.
The government, being made up of American citizens (in all three branches), as explained, has, ipso facto, free speech rights.
Now, please tell me - as it is your turn to do so - if you are a gentleman, how these Americans on US soil, who happen to govern (make the laws, judge the laws, execute the laws) these United States on any possible level (federal, state, county, municipal, etc.) could possibly NOT have free speech rights PURELY because they happen to govern? What is it that you imagine the government to be in the first place? Thank you.