The photographer mentioned in the article has a few videos up that illustrate this much better than the text https://www.youtube.com/watch?v=ff7nltdBCHs (for the "squinch" one).
The photographer stresses slightly raising one's lower eyelids (what he calls "squinching" -- a slight form of squinting), but if you notice, most of the examples he gives show people also slightly smiling more when they do this.
It's a bit hard to describe but you gotta look at yourself in the mirror and just try it for a while. After a few moments your brain will figure it out. Took me around 15 minutes to get my lower eyelids to move, but now after some days of practicing I can even do it with a single eyelid only!
To me this looks like the difference between "real" and "fake" smile (smiling with the eyes - or predator grin vs friendly smile). I have a friend who's a theathre actress - she smiles a lot wider than these models, but the eyes behave similarly.