You can achieve $\lvert F\rvert = 2\lceil\log_2 n\rceil$ by using all subsets of the form $\{x\in X \vert i^{\text{th}}\text{ bit of }x\text{ is }j\}$ for $i \in \{0,1,\ldots,\lceil\log_2 n\rceil\}$ and $j\in\{0,1\}$.

This rate is within a factor of two of best possible because there are at most $2^{\lvert F\rvert}$ different intersections you can form from $\lvert F\rvert$ sets. You require that all $n$ singletons be among this list of intersections, so $\lvert F\rvert \geq \log_2 n$.

Are you interested in the actual minimal value of $\lvert F\rvert$, or just asymptotics?

You can achieve $\lvert F\rvert = 2\lceil\log_2 n\rceil$ by using all subsets of the form $\{x\in X \vert i^{\text{th}}\text{ bit of }x\text{ is }j\}$ for $i \in \{0,1,\ldots,\lceil\log_2 n\rceil-1\}$ and $j\in\{0,1\}$.

This rate is within a factor of two of best possible because there are at most $2^{\lvert F\rvert}$ different intersections you can form from $\lvert F\rvert$ sets. You require that all $n$ singletons be among this list of intersections, so $\lvert F\rvert \geq \log_2 n$.

Are you interested in the actual minimal value of $\lvert F\rvert$, or just asymptotics?