The Stoics are relatively innocent and typically use logic properly for their reasoning as they have learned from Aristoteles except that some basic premises of the ancient Stoics are most likely false. There is no "Stoic logic" to refute.
I will have to come back to this too, but as to whether there is a "Stoic Logic":
QuoteStoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece. It was one of the two great systems of logic in the classical world. It was largely built and shaped by Chrysippus, the third head of the Stoic school in the 3rd-century BCE. Chrysippus's logic differed from Aristotle's term logic because it was based on the analysis of propositions rather than terms. The smallest unit in Stoic logic is an assertible (the Stoic equivalent of a proposition) which is the content of a statement such as "it is day". Assertibles have a truth-value such that they are only true or false depending on when it was expressed (e.g. the assertible "it is night" will only be true if it is true that it is night). [1] In contrast, Aristollean propositions strongly affirm or deny a predicate of a subject and seek to have its truth validated or falsified. Compound assertibles can be built up from simple ones through the use of logical connectives. The resulting syllogistic was grounded on five basic indemonstrable arguments to which all other syllogisms were claimed to be reducible.