Let $S=[0,1]\times [0,1]$ be the closed unit square. Suppose we label its four edges in cyclic order as $E_1,E_2,E_3,E_4$ so that $E_1$ is parallel to $E_3$ and $E_2$ is parallel to $E_4$.
Now, choose some partition $\{A,B\}$ of $S$. Must at least one of the following alternatives hold?
There is a quasicomponent of $A$ intersecting both $E_1$ and $E_3$.
There is a quasicomponent of $B$ intersecting both $E_2$ and $E_4$.
This statement is analogous to the (discrete) statement that a game of hex always has a winner. It seems that the continuous statement I'm asking about is intuitively correct (especially if we were to add a requirement that $A$ is closed or otherwise well-behaved), but it seems unclear how to handle the generality of this statement. Moreover, things do seem to break down: this question shows that both conditions could hold, as well as that neither could hold if we replace "quasiconnected" by "path connected". The comments give a counterexample for "connected" as well.
I'd primarily like to apply this in the case where $A$ and $B$ are reasonably well behaved (something comparable with $A$ being closed), but I wasn't able to quickly cook up any counterexamples even if we try to bring the axiom of choice into this, so I've been thinking that maybe such assumptions are unnecessary.