Is it possible to find an inverse system for this difference equation?
Just think of your system in simpler terms. Assume \$x[2n + 1] = a\$ and \$x[2n − 1] = b\$. Therefore:

$$
y = a+b
$$

Can you find a unique solution for both \$a\$ and \$b\$, knowing only the value \$y\$ ? The answer is no, there are infinte possible solutions. However, if you absolutely must find an inverse, constraints could be added to the problem, so that we have new probabilistic possibilities. 

Are \$x[2n + 1]\$ and \$x[2n - 1]\$ completely independent from each other, or is there some correlation? Is \$x\$ limited in amplitude or bandwidth? Perhaps a stochastic process can describe it? If enough constraints can be brought to the problem, one could then find an "expected" value for \$a\$ and \$b\$.

For example, say \$x\$ is described by a gaussian stochastic process, with mean \$\mu\$. Then the expected answer would be the one which brings both \$a\$ and \$b\$ closest to \$\mu\$.

One must realize that the system must then be solved for all \$n\$ in your considered range (do you wish to find solution \$\forall n \geq 0\$ ?). In this particular problem, this means solving:

$$
y[0] = x[1] + x[-1]\\
y[1] = x[3] + x[1]\\
y[2] = x[5] + x[3] \\
\dots
$$

If by some constraint or known value, you can find \$x[-1]\$, then you can solve for \$x[1]\$ and \$x[3]\$ and all others \$x[2n+1]\$.

In sum: is the system invertible? In strict terms, no. Is it possible to find an expected inverse signal? Given sufficient constraints, yes.