Don't they have this right under the "Commerce Clause". [which is indeed known to have been abused, but still]
Your opinion, publicly stated, might negatively affect commerce, which could have ripple effects in the economy of another state. Ergo, your ability to state your opinion publicly is regulable under the commerce clause. Don't worry: you're free to express your ideas in your mind, so long as you do not communicate them to anyone else in any form.
This line of reasoning is consistent with the Supreme Court's ruling in Gonzalez v Raich, Wickard v Filburn, et al.
The Commerce Clause has been blatantly twisted into an unconstitutional interpretation. They baldly lie and say it means something it clearly does not (remember, those decisions I cited defined commerce as including "not commerce"). Once you assert B AND NOT B == TRUE, then you can apply this logical fault to reason to any conclusion you wish.
So, to answer your question: yes, they assert they have this power under the constitution (technically, it's improper to say the government has "rights").
It might even be one of those rare constitutional applications of federal enumerated powers if it were limited to interstate commerce, but we all know that's not the case.