jmc

The absolute value of a number equals 1 if and only if the number equals either $-1$ or 1. Setting $2-|x|$ equal to 1 and to $-1$, we solve $$\begin{array}{rl}2-|x|=1& \text{or}2-|x|=-1\\ |x|=1& \text{or}|x|=3\\ x=\pm 1& \text{or}x=\pm 3.\end{array}$$ The sum of the squares of these four solutions is $(-1{)}^2+1^2+(-3{)}^2+3^2=\boxed{20}$.