Characterization of basically disconnected spaces We say that a topological space $X$ is basically disconnected if any cozero set of $X$ has open closure. I've proven that this is equivalent to the property that any disjoint cozero set $U$ and open set $V$ have disjoint closures.
Similarly, an extremally disconnected space is equivalent to any two disjoint open sets having disjoint closures.
Is the property of being basically disconnected space equivalent to any two disjoint cozero sets having disjoint closures? The proofs of above facts don't seem to work: we'd have to have that for a cozero set $U$, the set $\overline{U}$ is a zero set.

First of all, it should be noted that it does not make much sense to consider cozero sets in non-completely regular spaces. Therefore let us assume that all spaces are completely regular, T1.
Recall the following standard definitions for a completely regular, T1 topological space $X$:

*

*$X$ is basically disconnected, as defined above

*$X$ is F-space, if disjoint cozero sets are completely separated

*$X$ is F$^\prime$-space, if disjoint cozero sets have disjoint closures
(i.e., the condition the OP is asking for)

It is easy to see that 
$X$ basically disconneted $\Rightarrow$
$X$ (strongly) zero-dimensional and F-space $\Rightarrow$
$X$ F-space  $\Rightarrow$
$X$ F$^\prime$-space. 
It is well-known that none of these implications is reversible.
See, for instance, here and the references given in this paper. It also contains an example of locally compact F$^\prime$-space, which is not F-space.
A perhaps more striking counterexample to the OP's question is the Stone-Cech compactification
$\beta \mathbb R^+ \setminus\mathbb R^+$ of the non-negative reals $\mathbb R^+$, which is a connected F-space, hence not basically disonnected.
(See Gillman, Jerison, Rings of continuous functions, 14.27.)
