'wanted' or 'had wanted'

The past perfect (had + past participle) is mandatory when there is no other information in the sentence to make the sequence of events clear. For example:

When I arrived at the party, he had left.

Here the past perfect clearly indicates that the leaving happened before the arriving. If this is the case, then you cannot use the preterite:

When I arrived, he left

because this reverses the meaning: the leaving happened after the arriving.

In the sentence Ukrainians wanted to join the EU even before Russia occupied the Crimea and Eastern Ukraine, the word before is sufficient to disambiguate the sequence of events, so while the past perfect is grammatical, it is not needed.

Indeed, the past perfect is the marked variant here and, as Erik Kowal correctly points out, it suggests an implicature missing from the preterite version.

Both variants are acceptable, but they carry different implications.

Ukrainians wanted to join the EU even before Russia occupied the Crimea and eastern Ukraine

is a straightforward statement of fact.

Ukrainians had wanted to join the EU even before Russia occupied the Crimea and eastern Ukraine

conveys the same information, but likely also suggests something further in connection with the Ukrainians' wish to join the EU – specifically, because the pluperfect form 'had wanted' implies a change in some aspect of their desire.

For instance:

Ukrainians had wanted to join the EU even before Russia occupied the Crimea and eastern Ukraine. Russia's occupation of their eastern borderlands only strengthened their ambition to join.

NB: Note that today, it is usual to omit the the from the name of the country (simply 'Ukraine' rather than 'the Ukraine'). Also, 'eastern Ukraine' is a geographical description meaning simply 'the eastern part of the country'; 'Eastern Ukraine' implies that there is a political entity called by that name.