Let $G$ be a finite group and $n$ be a positive integer. Define 
$$X_n(G):=\{x\in G: x^n=1\}.$$
Define $$J_n=\left\{\frac{|X_n(G)|}{|G|}: G\,\text{is a finite group}\right\}.$$
**Question.** Does there exist a positive integer $n$, such that 
$J_n=\mathbb Q\cap[0,1]$?

Here $\mathbb Q$ is the set of rational numbers? The answer to the simpler question below is also desirable to me: 

Does there exist $n\in\mathbb N$, and a family of finite groups $(G_n)_n$ such that 

 1. $|X_n(G_k)|<|G_k|$ for all $k$, 
 2. $0<\prod_{k=1}^\infty\frac{|X_n(G_k)|}{|G_k|}$?