I could not prove that "Every Boolean ring is commutative", but I found it on the Internet. I am giving the idea of the proof that I "learned".

Let $R$ be the Boolean ring.

We can easily prove that $x+x = 0 \; \forall \; x \in R$. Then, we again use the same idea that $(x+y)^2 = (x+y), \; \forall \; x,y \in R$ to get that $xy=-yx$ and use the fact that $xy= -xy$ to get the result.

We had to find the new property for the Boolean ring that $x+x =0 $ to solve that it is commutative. Even with this solution, I cannot understand what makes 
a Boolean ring commutative.
Is there any intuitive way to understand that Boolean ring is commutative? I am not looking for just a computational answer (if possible, there should be some idea behind the computation). I am hoping for any sort of intuition or idea (mathematical or in plain English) that tells us why Boolean ring is commutative. 

P.S.: A Boolean ring $R$ is a ring in which $a^2 = a, \; \forall \; a \in R $.