The correct construction for a topological category is as follows:
If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.
Now it suffices to give the answer for simplicial categories.
However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.
The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.
Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$
Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:
$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.
for any simplicial category $C$.
Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).
As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...
Edit: Alright, so the reason why they agree is covered in ยง4.2.4 of HTT, I'm pretty sure.
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.
Best Answer
For $\def\B{{\rm B}} \def\bB{{\bf B}} \def\Spin{{\rm Spin}} \def\String{{\rm String}} \B\Spin(n)$, simply equip the $n$-planes with a spin structure, as originally proposed by Stolz and Teichner.
For $\B\String(n)$, equip the $n$-planes with a string structure, as described in a paper by Douglas and Henriques.
Also, I am not sure what the intended difference between $\B$ and $\bB$ is, but if $\bB$ does refer to the stack version of these classifying spaces, then the above constructions continue to work perfectly well for stacks: to a smooth manifold $S$ assign the groupoid respectively 2-groupoid of vector bundles with base $S$ and whose fibers are equipped with a spin respectively string structure.
This yields the stacks $\bB\Spin(n)$ and $\bB\String(n)$, and applying the shape functor to these stacks yields the classifying spaces $\B\Spin(n)$ and $\B\String(n)$.