I had this question myself recently and searched for answers. Here's a quick summary of findings:
Here's Wikipedia's breakdown of occular nerve mapping. The most important distinction is voluntary vs. involuntary. An example of the latter is "pupil dilation" – difficult to control, subject to common stimuli. An example of the former is "direction of gaze." Basically, greater affinity for voluntary affectation = more complex heuristics for analysis = fewer easy rules like "looking up is visual."
My opinion is that most eye-direction articles are 99% pseudo, pop content. I suppose mind-reading intrigues us. Most related university studies I read fell under "eye-tracking" and "point of gaze" research, but the left-brain right-brain visual/auditory/kinesthetic lingo was nowhere to be found in relevant, recent literature.
Speaking of VAK, I'd argue learning styles theory is a fair comparison. It has been disproved many times over yet persists as a dominant theory of ed psych. Even the article you linked alludes to it:
"[Looking up] could indicate that they need you to demonstrate, rather than verbally explain."
Anyway, I don't want to beat a dead horse. Bottom line: according to my research, it's mostly crap.