No, let $\mathcal F_n$ be the powerset of $\{0,1\}$ and let $B$ be the event that $X_n=1$ for at most finitely many $n$. Then if $B$ is 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.