As I'm sure you know, there are a lot of variations on how quantiles are calculated in various software. The 25th percentile, e.g., doesn't always line up with a value in the dataset, so sometimes nearest rank methods are used, otherwise a linearly interpolated data point, where interpolation is done in various ways.

In any event, none of these methods assume normality, or rely on CDFs of a normal curve.

If they did, every box plot would be symmetric.

The fact some people think that boxplots are constructed in such a way is a pretty good reason to take the author's article seriously as for how boxplots are confusing.