I need to define random variables such that their E(X) = 1 and Var(X) = 1 and these values need to be non-negative. So far, the only assignment of values to a random variable that I can think of that satisfy these criteria are for example X = {0,2}, X = {0,0,2,2}, or X = {0,0,0,2,2,2} and so on.
I was wondering if there are other such assignments that I am unable to think of that have E(X) = 1 and Var(X) = 1.
This describes what is implicit in some of the comments, and in a deleted answer.
Consider the set of points $S=\{(x,x^2):x\ge0\}$ in the plane. Any probability distribution supported on $S$ whose center of mass is the point $(1,2)$ will supply an example of the sort the OP wants.
(What's special about $(1,2)$? Because it is $(EX,EX^2)=(EX,\text{Var}(X)+(EX)^2)=(1,1+1)$.)
For instance, $(1,2)$ is the weighted average of the endpoints of any chord of $S$ passing through $(1,2)$. For instance, the OP's original examples correspond to placing equal weights at $(0,0)$ and $(2,4)$; and @lulu's comment gives another example of this form.