1. **Problem:** Find the intersection of sets $A = \{x \mid x \leq 0 \text{ or } x > 1\}$ and $B = \{-2,0,1,2\}$.\n\n2. **Formula and rules:** The intersection $A \cap B$ contains elements that are in both $A$ and $B$. We check each element of $B$ to see if it satisfies the condition for $A$.\n\n3. **Check elements:**\n- $-2$: Since $-2 \leq 0$, $-2 \in A$.\n- $0$: Since $0 \leq 0$, $0 \in A$.\n- $1$: Since $1 \leq 0$ is false and $1 > 1$ is false, $1 \notin A$.\n- $2$: Since $2 > 1$, $2 \in A$.\n\n4. **Conclusion:** The intersection is $\{-2, 0, 2\}$.\n\n**Final answer:** C. $\{-2, 0, 2\}$