No, let $\mathcal F_n$ be the powerset of $\{0,1\}$.
Let us write $X\in\{0,1\}^{\mathbb N}$ as $(X_n)_n$.
Let $B=\{X:X_n=1$ for at most finitely many $n\}$. Suppose $B=$ lim sup $A_n$. Then
$X_n=1$ for at most finitely many $n$ $\iff$ infinitely many $A_n$ occur.
Case 1: for infinitely many $n$, $A_n=\{0,1\}$. Then $X_n=1$ for all $n$ is a counterexample.
I'll let you think about the other cases.