Suppose that we have two very nice CDF's $A$ and $B$ both are continuous and has support $[0,1]$, and pdf's $a$ and $b$ respectively.
Let's say that the two CDF's are related in a way that
If $A(x)\leq \frac{1}{2}$, then we have $\int^1_0(x-y)^2\big[\frac{1}{2}-B(y)\big]b(y)dy\leq 0$ and
If $A(x)\geq \frac{1}{2}$, then we have $\int^1_0(x-y)^2\big[\frac{1}{2}-B(y)\big]b(y)dy\geq0$.
What kind of relationship between $A$ and $B$ will satisfy those inequalities?
(My guess is that the two CDF's are close to or similar to each other.. for example, if they both are CDF's of Uniform dist. then it satisfies the inequalities.) but I'm not sure exactly what would be the exact relationship..)
Let $X\sim A,\; Y\sim B$, then your equations are saying:
$$\textrm{sgn}\left(A(X)-\frac{1}{2}\right)\cdot E\left[\left.(X-Y)^2\left[\frac{1}{2}-B(Y)\right]\right|X\right]\geq 0$$
This is not a simple relationship to state in some sort of intuitive language.