Is "Not X is Not Required" an example of a double negative?

This has little to do with language and lots to do with logic. There are four, and only four, logical possibilities. I use R to represent Required and {} to represent Not:

X R ; X{R} ; {X}R ; {X}{R}

Each is different from the others.

Your opinion is therefore correct that "Not X is Not Required" is very different to "X is Not Required".

Or, more formally: {X}{R} ≠ X{R}

This is just one example of the differences between the four possibilities.

I add that it is also clear from this logic that (X not required) does not imply (X Required); that would be to ignore the other two logical possibilities.

Your question title asks if {X}{R} is a double negative; it contains two negatives so is a double negative, but this seems a trivial point compared to the above.

It is true that there are two negations here and that it is possible to eliminate both of them (and so get a clearer, more natural formulation), but they do not simply cancel out. To see what this sentence actually means, consider first that

Not X is not required.

is equivalent to

It is not required to not X.

A more natural way of saying to not X (where X is some action) is to abstain from X. We thus get

It is not required to abstain from X.

Now, to be required to abstain from something is to be prohibited from doing it. The sentence can therefore be transformed into

It is not prohibited to X.

What is it to not be prohibited to do something? It is, of course, to be permitted to do it. So we get

It is permitted to X.

which is a simple, clear reformulation of the sentence we have started from.

The formal, symbolic apparatus of deontic logic makes such equivalences more obvious than they are in a natural language, such as English.