He signed a lease that specified what he could have on his balcony. Nobody forced him to sign the lease. If the counterparty landlord wants the flags removed, he should remove them. That's what freedom of contract is all about; one undertakes obligations in order to receive something. The vet has a leased apartment, in exchange for which he agreed to refrain from certain acts. He's violating his promise. This has absolutely nothing to do with freedom of speech, which can only be abridged by a government, not in a private contract between two consenting parties.

This is boss trying to prove something. Leave the vet and his flags alone. He is being bullied.