I'm not sure how to interpret this problem.
Find a domain for the qunatifiers in: $$\exists{x} \exists{y}(x\neq y \wedge \forall{z}((z=y)\ \lor(z=x))) $$ such that this statement is false.
So, the translation to English would be: There exists some $x$ and some $y$ that are not the same and $z$ is $y$ or $z$ is $x$.
It's gibberish to me at this point.
There are two objects, $x$ and $y$. Since mathematical objects, even under different labels, need not be distinct, we also require $x\neq y$. So there are two distinct objects called $x$ and $y$.
And for every object $z$ that we take, either $x=z$ or $z=y$. So any object in the universe is actually either $x$ or $y$ (but not both, as these are distinct objects).
Now consider the example "I have one orange and one apple, and any fruit I have is either an orange or an apple". What have I told you? I told you that all my fruits are these two, an orange and an apple. But if I also have a peach then I was lying when I told you that. Because I have a fruit which is neither the apple nor the orange.
The sentence you are given is no different, only it doesn't care for oranges and apples and other things. It is an abstract claim about an abstract universe. When will it be true? When there are exactly two objects in the universe, and it will be false if there is a third object.