Every element in $x \in A$ in a commutative bolean ring $A$ is by definition equal to $xx$, and so therefore there are no irreducible elements in the usual sense. So I ask, how might one define irreducibility for boolean rings?
For instance, a non-unit $x \in A$ is is irreducible if it can't be written $xz$ for any $z \neq x, z \in A$ a non-unit.
In domains, irreducibility of $x$ corresponds to $(x)$ being maximal in the poset of proper principal ideals.
So, you could use that as the definition.