As Michael wrote, we may assume $c=1$.
Your equation can be written $diag(\phi(X))=[1]$ where $\phi$ is the linear function $\sum_{k=0}^{\infty}A^k\otimes A^k=\sum_{k=0}^{\infty}(A\otimes A)^k$ (Here we stack the matrices row by row). This series converge for every $X$ iff $\rho(A\otimes A)<1$, that is, iff $\rho(A)<1$; in the sequel, we assume that the previous condition is fulfilled. Note that your condition about the $(A_{i,j})$ is not sufficient, except perhaps if $c$ is small;
We obtain $diag((I_{m^2}-A\otimes A)^{-1}X)=[1]$. The general solution is $X=(I_{m^2}-A\otimes A)U$ where $U$ is an arbitrary $m\times m$ matrix with diagonal $1$. In particular, $X$ depend on $m^2-m$ free parameters ($X$ has $m^2-m$ degrees of freedom).
EDIT 1. I just see that $X=diag(x_i)$ is diagonal. Moreover, I obtain the same equation than Michael.
EDIT 2. It is better to reason as follows: let $B=[b_{i,j}]=(I_{m^2}-A\otimes A)^{-1}$. We obtain explicitly $X$ as a function of the $(b_{i,i})$: for every $i$, $x_i=1/b_{i,i}$.
In particular, a necessary condition is that, for every $i$, $b_{i,i}\not=0$ (and there is a sole solution in $X$), otherwise there are no solutions in $X$.