$B(z)+B(-z) = 2c$, explain the structure of $b[n]$ and find the constraint of its length given that $c$ cannot be $0$.
This is a homework problem. "Explain the structure" means that $b[n]$ is zero for certain values of $n$, and has a certain shape.
I'm trying to take the inverse $\mathcal Z$-transform of $B(z)$ and $B(-z)$, but I'm not sure how the inverse $\mathcal Z$-transform of $B(-z)$ is related to $b[n]$, so I'm stuck...can anyone give me some advice?
Just use the definition of the $\mathcal{Z}$-transform:
$$B(z)=\sum_{n=-\infty}^{\infty}b[n]z^{-n}\tag{1}$$
from which it follows that
$$B(-z)=\sum_{n=-\infty}^{\infty}b[n](-z)^{-n}=\sum_{n=-\infty}^{\infty}b[n](-1)^nz^{-n}\tag{2}$$
From $(2)$, the sequence that corresponds to $B(-z)$ is $b[n](-1)^n$. Now taking the inverse $\mathcal{Z}$-transform of
$$B(z)+B(-z)=2c\tag{3}$$
gives
$$b[n]+b[n](-1)^n=2c\delta[n]\tag{4}$$
For odd $n$, the left side of $(4)$ equals zero. For even $n$ it equals $2b[n]$. Consequently, equation $(4)$ can only be satisfied if $b[n]=0$ for even $n$ except for $n=0$, where we require $b[0]=c$:
$$b[n]=\begin{cases}c,&n=0\\ 0,&n\text{ even}\end{cases}$$
