While this may qualify as "negative thinking," whatever that means, it would be misleading to extrapolate this example to mean Stoicism is about negative thinking. The context of such a statement has to do with a key tenet, that "happiness" (eudaimonia), or a good flow of life, derives from the rational selection of the primary things according to nature. The "primary things" refer to the things you need to live, to survive. So if you choose rationally and in a manner that agrees with nature (not Nature, meaning the mountains and the trees, but the inherent order of the universe), you should be fine. Why? For the Stoics, if you can maintain your life, only your virtue matters. The rest means nothing, because you recognize that you have no control over it, over this ever-changing world. This attitude is illustrated in the story: a man runs up to a Stoic and announces sorrowfully, "Your son is dead!" The Stoic responds, "Yes, I knew my son was mortal." So the "negative thinking" in Stoicism usually involves recognizing the nature of the world: bad things could happen, so don't be surprised or emotionally affected. They cannot harm your virtue.

As for the technique of emotionally conditioning, I'm not sure, but it looks analogous to Aristotle's technique for correcting character in the Nicomachean Ethics: if you are too stingy, spend overly generously and eventually you'll find the golden mean. If you're too generous, be stingy for a while and you'll train your nature to get used to the middle ground. So the example mentioned in the WSJ seems to the technique adapted to address fear. The underlying Stoic assumption is that poverty is nothing to be feared, because it does not interfere with your "happiness"

An American soldier survived being a POW in Vietnam by adhering to Stoicism. Look up James Stockdale.

I'll take a look at the article when I have more time, but I just want to point out what I suspect to be a misrepresentation, or at least a simplification, of Stoicism.