Why is this function not invertible?
The principle here is that you can't get information from nothing. If a function throws away information, the inverse function would need to magically reproduce it. In this case, your function is throwing away the sign of the input value. Let's look at two examples. In the first, x[n] = 1 for all values of n:

$$x[n-1] = 1$$
$$y[n] = \frac 1 {x[n-1]^2} = \frac 1 {(1)^2} = 1$$

In the second example, x[n] = -1:

$$x[n-1] = -1$$
$$y[n] = \frac 1 {x[n-1]^2} = \frac 1 {(-1)^2} = 1$$

In both cases, y[n] = 1.

Now, imagine that you're told y[n] = 1, and you want to find x[n]. How can you tell which of the two example sequences to choose? You can't. The difference between them is the sign, but squaring the input gets rid of the sign! Thus, the function is not invertible.