Original Problem: Counterexample given below by user francis-jamet.
Let $A\subset \mathbb Z_n$ for some $n\in \mathbb{N}$.
If $A-A=\mathbb Z_n$, then $0\in A+A+A$
New Problem: Is the following statement true? If not, please give a counterexample.
If $A-A=\mathbb Z_n$ and $0\not\in A+A$, then $0\in A+A+A$.
For the original problem, there is a counterexample for $n=24$ and $A=\{3,9,11,15,20,21,23\}$.
There are no counterexamples for $n \leq 23$.
For the new problem, there is a counterexample:
$n=29$ and $A=\{4,5,6,9,13,22,28\}$.
There are no counterexamples for $n \leq 28$.