Eternal recurrance of the same implies a fixed unchanging essence, but Buddhist ontology is clear that things lack unchanging self-nature, and holds that everything is impermanent. See the Brahmanjala sutra (section 3, part 1) for the argument against any kind of eternalism.
Note that for Mahayana Buddhism (broadly, Tibetan & Chinese), personal enlightenment is foresworn until every being is enlightened, this is called the bodhisattva path. Comparison here might be drawn with free embrace of nearly-endless recurrance. There are various theological subtleties on this in the Mahayana tradition, along the lines that it is delusion which ceases or is extinguished rather than all existence ceasing, but there are conflicting interpretations (karmic causation ceases at awakening, but previous karma remains to be dealt with).
What happens to a Buddha after death, and so to all beings for Mahayana Buddhists on completion of the bodhisattva path, is one of the 'unexpounded questions'. For everyone else they will be recur not as the same being (reincarnation), but as beings inheriting their karma, like a candle being lit by another candle. We replay our conflicts and tensions until we resolve them, or are reconciled to them.
Heidigger points out that in Nietzsche's first mention of eternal recurrance in The Gay Science:
"The way Nietzsche here patterns the first communication of the thought of the 'greatest burden' [of eternal recurrence] makes it clear that this 'thought of thoughts' is at the same time 'the most burdensome thought'"
I would argue that Buddha's teachings often use thought experiments, like the parable of the mustard seed, and the parable of the poison arrow. Metaphorical, rather than literal truths. The Buddha took the cultural furniture of his era, reincarnation concerned with a soul's karma, and reworked it into a doctrine of rebirth concerned with rebirth of causes and conditions in a situation of no-self, anatta. Mahayana Buddhism took that further, to concern for causes and conditions of all beings. This can be seen as useful to contemplate in the way the thought-experiments are. And Nietzsche's eternal return properly used, is also valued for it's effect in contemplating it, rather than literal truth.
Now, then people of a physicalist-materialist bent say, but which and what can be literally true? Rebirth? Recurrance of a reconciled being, a boddhisattva? Modern physics is contending with the idea of Many Worlds, of a higher dimensional probability landscape. Buddhists hold that there have been multiple universes, and that karma and minds have been around infinitely long. Could that be an intuition of a larger view of reality and causality? I think there is space in understanding of consciousness through the Penrose-Hameroff model of consciousness as occurring at a quantum level, for recignising that we are companions with, shaped and reflected by, all our possible selves. In this sense there may be a larger truth reconciling Buddhist thought and science, which we will I think we will need techniques from both to understand.
You might find this article helpful, which draws a comparison between Nietzsche and one of Buddhism's deepest thinkers Nagarjuna, in their philosophical techniques.