Oh! Well, that turns it from some complex mass of edge cases into a simple proposition.
If you go by the completely psychotic RAW or the Wizards FAQed answer on natural+flurry, then the flurry would not drain but any additional natural attacks would. Of course you'd have to get additional natural attacks, which just being a human soul eater or whatnot wouldn't get you.
If you go by the more reasonable interpretation accepted as the answer on the other question, and that's how I'd rule, then I'd also rule that sure, it's a flurry but "with" natural weapons and so totally, the energy drain would proc on each. It says "touch" after all, and considering an unarmed attack not a "touch" stretches in-game-world credibility.