For the symmetric group $\Sigma_n$, you can take
\begin{align*} 
 E\Sigma_n &= \{\text{injective functions } \{1,\dotsc,n\}\to\mathbb{R}^\infty \} \\\\
 B\Sigma_n &= \{\text{subsets of size $n$ in } \mathbb{R}^\infty \}
\end{align*}

Now let $G_n$ be the group of braids on $n$ strings, and let $H_n$ be the subgroup of pure braids.  We have
\begin{align*} 
 BH_n &= \{\text{injective functions } \{1,\dotsc,n\}\to\mathbb{R}^2 \} \\\\
 BG_n &= \{\text{subsets of size $n$ in } \mathbb{R}^2 \}
\end{align*}
These spaces have trivial homotopy groups $\pi_{k}(X)$ for $k\\geq 2$, so 
$$ EH_n=EG_n= \text{ universal cover of } BH_n = 
     \text{ universal cover of } EH_n.
$$
I think I see a proof that this space is homeomorphic to $\mathbb{R}^{2n}$, but I don't know if that is in the literature.