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The value $0$ always occurs infinitely often regardless of the value $n>1$. To see this, consider the mapping $L_n:\mathbb{Z}_n^2\rightarrow\mathbb{Z}_n^2$ defined by $L_n(x,y)=(y,x+y)$. Then the mapping $L_n$ is invertible. Then if we set $F_{0,n}=[0]_n,F_{1,n}=[1]_n$ and $F_{i+2,n}=F_{i+1,n}+F_{i,n}$, then $F_{i,n}$ is the $i$-th Fibonacci number modulo $n$. On the other hand, $L_n(F_{i,n},F_{i+1,n})=L_n(F_{i+1,n},F_{i+2,n})$, so $L_n^m(F_{i,n},F_{i+1,n})=(F_{i+m,n},F_{i+m+1,n})$ for all $m\geq 0$. Since $L_n$ is invertible, there must be some $m>0$ where $([0]_n,[1]_n)=L^m_n([0]_n,[1]_n)=L^m_n(F_{0,n},F_{1,n})=(F_{m,n},F_{m+1,n})$, so $F_{m,n}=[0]_n$.