Question
I need a division algebra D of characteristic 2, of 4 dimensional over its center Z(D), with an element $x\not \in Z(D), x^2\in Z(D)$. Is there any such D?
What I know
The quaternion algebra does not work in a field of characteristic 2. And I know that any finite division rings are fields.
Quaternion algebras do work in characteristic $2$, they are just a little different (though it is possible to give a unified definition). Let $K$ be a field of characteristic $2$, and let $a\in K$ and $b\in K^\times$.
Then you can define a $K$-algebra $Q$ with generators $i$ and $j$ satisfying $$i^2+i=a,\quad j^2=b,\quad ij = j(i+1).$$
The algebra $Q$ is central simple over $K$, of dimension $4$ (with basis $1,i,j,ij$). And you see that $j^2$ is in the center. Also, $Q$ is usually a division algebra, unless $b$ is a norm for $K(i)/K$.
Where does this definition come from? Over a field of characteristic $2$, there are $2$ types of quadratic extensions: separable ones, and inseparable ones. The inseparable quadratic extensions $L/K$ have the familiar form $L\simeq K(\sqrt{b})$ for some $b\in K^\times$. But the separable ones look like $L\simeq K(\wp^{-1}(a))$ for some $a\in K$, where $\wp: K\to K$ is the function $\wp(x)=x^2+x$.
Then in the definition of $Q$, we see the two types of quadratic extensions: $K(j)\simeq K(\sqrt{b})$ is inseparable, and $K(i)\simeq K(\wp^{-1}(a))$ is separable. Note that $K(i)/K$ is a Galois extension, and the conjugation by $j$ in $Q$ induces the non-trivial automorphism of $K(i)/K$, ie $i\mapsto i+1$.