. A set $X$ of boys is separated from a set $Y$ of girls if no boy in $X$ is an acquaintance of a girl in $Y$. Similarly, a set $Y$ of girls is separated from a set $X$ of boys if no girl in $Y$ is an acquaintance of a boy in $X$. Since acquaintance is assumed mutual, separation is symmetric: $X$ is separated from $Y$ if and only if $Y$ is separated from $X$.

This enables doubly counting the number $n$ of ordered pairs $(X, Y)$ of separated sets $X$, of boys, and $Y$, of girls, and thereby showing that it is congruent modulo 2 to both numbers in question.

Given a set $X$ of boys, let $Y_{X}$ be the largest set of girls separated from $X$, to deduce that $X$ is separated from exactly $2^{\left|Y_{X}\right|}$ sets of girls. Consequently, $n=\sum_{X} 2^{\left|Y_{X}\right|}$ which is clearly congruent modulo 2 to the number of covering sets of boys.

Mutatis mutandis, the argument applies to show $n$ congruent modulo 2 to the number of covering sets of girls.

Remark. The argument in this solution translates verbatim in terms of the adjacency matrix of the associated acquaintance graph.