Mathematical objects that satisfy commuting property and $A^2=0$ Is there a set of any numbers, matrices or their generalizations that satisfies all of the following?
1) $A^2=B^2=C^2=D^2... =0$ where $A,B,C,D..$ are unequal mathematical objects
2) Objects in the set commute.
3) Other products of objects that do not involve the squared number of an object are not zero.
Also, what would be the restriction on the number of objects?

In a group $G$, it can be easily shown that $1)\implies 2)$. But $2)\implies 3$ is wrong if the order of the group is odd. You can easily show that:
If $G$ is abelian of odd order, then the product of all elements of $G$ is the identity element. 
So now the case if $G$ is even is left. 
Hence (abelian) groups of odd order do not work in your case. 
