Embedding of exotic manifold

Not necessarily. Here's one such (non-compact) example.

There are uncountably many smooth structures on $\mathbb{R}^4$. An exotic $\mathbb{R}^4$ is said to be small if it can be smoothly embedded in the standard $\mathbb{R}^4$, and large otherwise; both small and large $\mathbb{R}^4$'s exist. A small $\mathbb{R}^4$ (or standard $\mathbb{R}^4$) embeds in $\mathbb{R}^4$, large $\mathbb{R}^4$'s do not. So the smallest $n$ for which a large $\mathbb{R}^4$ embeds into $\mathbb{R}^n$ satisfies $n > 4$ (and $n \leq 8$ by the Whitney Embedding Theorem). In fact, every large $\mathbb{R}^4$ embeds in $\mathbb{R}^5$, see this question.

Every exotic sphere (of dimension $\ge 7$) is an example. Indeed, suppose $\Sigma$ is an exotic $n$-dimensional sphere and that it embeds (smoothly) in $R^{n+1}$. Then it bounds a contractible compact submanifold $W$ of $R^{n+1}$. Now, remove a small ball $B$ from $W$. The result is an h-cobordism between $\Sigma$ and the boundary of $B$, which is the usual sphere $S^n$. Hence, by the smooth h-coboridism theorem, $\Sigma$ is diffeomorphic to $S^n$.