Negative Skewness definition Negative Skewness is defined as:
The left tail is longer; the mass of the distribution is concentrated on the right of the figure. 
Is it possible to obtain an Negative Skewness distribution with both positive or negative values for the difference between the 75th and 25th percentiles: $q_{75} - q_{25}$ can be >0 or <0?
If so, why is it used the "negative" word for this?
 A: 
Negative Skewness is defined as: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. 

I'd be inclined to say that's more a description that an actual definition; we'd have to make those concepts more precise to have it really count as the definition of skewness.

Is it possible to obtain an Negative Skewness distribution with both positive or negative values for the difference between the 75th and 25th percentiles: $q_{75}−q_{25}$ ... can be >0 or <0?

If I understand your right, you seem to be confused there. The upper quartile is always at least as large as the lower quartile. That difference is the interquartile range -- which is not related to skewness at all, but to how "spread out" the distribution is (its scale). 
There is a measure of skewness based on the quartiles, the quartile skewness (though it has numerous other names, including Bowley skewness and Galton skewness):
$$\frac{Q_3-Q_2-(Q_2-Q_1)}{Q_3-Q_1}=\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1}$$
(here $Q_1$ is the lower quartile, your $q_{25}$, and so on).
Personally, I tend to think of it as "boxplot skewness" because it's effectively the impression of skewness you get just from looking at the middle part (the box) of a boxplot:

Note that this measure will be positive if the upper quartile is further above the median than the lower quartile is below it, and negative if the lower quartile is further away; this accords in a general sense with the original notion. The quartile skewness ignores the extreme tails and measures that focus on the tails further out may suggest skewness running in the opposite direction to the quartile skewness.
