No, you're totally off. We already have a federal regulatory agency, the EEOC, which is ostensibly there to protect employees from discrimination based on a host of things including ideology. That agency is so backed up with claims it takes months or years to work through the process. All Trump did was to say "stop it" to things that are already illegal unfair infringements on people's rights.
He says in his own tweet he's banning "people and companies" from doing business with the government. So I'm presuming a sole proprietor who historically provided services to a government entity that doesn't share Trump's ideology would be precluded from getting a contract. That's not a free speech issue?
(Perhaps I don't understand the policy itself as you've stated. Quite possible as I haven't read up on it.)