The facts we need here are

1. For each ideal $I$ of $A$, the quotient ring $A / I$ is also a Boolean ring. Because for every $i \in A / I$ we have a ring morphism $\varphi : A \to A / I$ and an element $a \in A$ such that $$\varphi(a) = i$$ To show $A / I$ is a Boolean ring, means for each $i \in A / I$ we have $i * i = i$. Now $$i * i = \varphi(a) * \varphi(a) = \varphi(a * a) = \varphi(a) = i$$ as desired.
2. If a Boolean ring $R$ is also an integral domain, it has only two elements. Because from $a \in R$ we have $$a * a = a$$ rewrite it to $$a * (a - 1) = 0$$ Now because $R$ is an integral domain, hence $a = 0 \lor (a - 1) = 0$ for all $a \in R$. Therefore, the only two elements are $0$ and $1$.

For each prime ideal $x$, the quotient ring $A / x$ is an integral domain, and hence has finite elements, which implies it's also a field, hence $x$ is a maximal ideal.