Let $f(u)$ be a polynomial of degree $m, m \geq 2,$ which splits into distinct linear factors over a finite field $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u]/\langle f(u)\rangle$ be a finite non-chain ring. In an earlier paper, we studied duadic and triadic codes over $\mathcal{R}$ and their Gray images. Here, we study duadic negacyclic codes of Type I and Type II over the ring $\mathcal{R}$, their extensions and their Gray images. As a consequence some self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over $\mathbb{F}_q$ are constructed. Some examples are also given to illustrate this.