A property of an ultrafilter

Here's a try for $n=3$ which I think generalizes to any $n$.

Well-order $\mathcal{F}$ as $\{F_\alpha\}$. By discarding any $F_\alpha$ which is contained in $\bigcup_{\beta <\alpha} F_\beta$, we can ensure that each $F_\alpha$ contains at least one point which is not in any previous $F_\beta$, without affecting $U = \bigcup F_\alpha$.

We can now inductively construct a set $V \subset U$ with the property that $|V \cap F_\alpha|= 1$ or $2$ for all alpha. At each $\alpha$ at least one element of $F_\alpha$ is new and we can include it or not to ensure the condition for $F_\alpha$.

Now $U \setminus V$ has the same property, so since $U \in \mathcal{U}$, wlog we can assume $V \in \mathcal{U}$.

Let $\mathcal{E}_1 \subseteq \mathcal{F}$ consist of those $F_\alpha$ which intersect $V$ in exactly one point, and let $\mathcal{E}_2$ consist of the other $F_\alpha$, which all intersect $V$ in two points. If the union of $\mathcal{E}_1$ belongs to $\mathcal{U}$ then we are done. Otherwise the union of $\mathcal{E}_2$ belongs to $\mathcal{U}$ and this reduces us to the $n=2$ case.

The general reduction turns the problem for an arbitrary $n$ into the same problem for some smaller $n$.

For $n=2$ and any $X$: for each $x\in \cup \mathcal F$ fix any edge $(x,y)\in \mathcal F$ and draw an arrow from $x$ to $y$. Remove all other edges. Remaining edges still have the same union as $\mathcal{F}$, but they form a directed graph with outdegrees at most 1. Such a graph has a proper 3-coloring (it suffices to prove this for finite subgraphs by compactness theorem, and for them the result is well known and easy: remove the vertex with minimal in-degree, it is at most 1, and proceed by induction). One of three colors works.

Consider this with assuming that $\mathcal{F}$ consists of finite subsets of bounded cardinal; let $n$ be the max of these cardinals and let us argue by induction on $n$. The goal is to find $\mathcal{E}\subset\mathcal{F}$ and $U\in\mathcal{U}$ such that $U\subset\bigcup\mathcal{E}$ and and $|E\cap U|\le 1$ for every $E\in\mathcal{E}$.

First, consider a maximal subset $\mathcal{E}$ of the cover such that no element is in the union of others (we call this ``minimal"). So $V:=\bigcup\mathcal{F}=\bigcup\mathcal{E}$. Write $V=V_1\cup V_2$, where $V_2$ is the set of elements of $V$ that belong to at least two elements of $\mathcal{E}$. The minimality of $\mathcal{E}$ implies that no $E\in\mathcal{E}$ is contained in $V_2$ (i.e., has nonempty intersection with $V_1$).

Since the intersection of elements of $\mathcal{E}$ with $V_1$ are pairwise equal or disjoint, we can partition $V_1$ into $n$ subsets each intersecting, each element of $\mathcal{E}$ in at most a singleton. Hence we can conclude if $V_1\in\mathcal{U}$ (with $U$ being one of those $n$ subsets of $V_1$).

Otherwise, $V_2\in\mathcal{U}$. Since $|E\cap V_2|\le n-1$ for all $E\in\mathcal{E}$, we can conclude by induction (and find $U\subset V_2$).

PS Nik Weaver posted an answer while I was writing this one, but I still post, although I guess it's the same idea.