Today, among us Muslims there are people who often glorify the thirteen centuries rule of "Islamic caliphate". This glorification has resulted in various nationalist discourse in 20th century and is still constructed today. It is commonly agreed that a triumphant past under the banner of "Islamic civilization" existed (and it is a fascinating subject to inquire by itself).
Prior to Muslims' rule in 7th century, however, most of Middle East was Christian. The Christians were under the domain of Roman and Persian, two huge civilization by its own. I know there were people like Cosmas Indicopleustas or Eusebius who see Roman Empire as a civilization so great that it is believed to last until the end of time. It didn't, however - and today this Eastern Mediterranean region has always been identified as a "Muslim region".
So has there ever been a collective/historical memory of pre-Muslim past among the Middle Eastern Christians? Is something like "Nestorian Christians were the precursor of modern science!" or a more confrontational "things were better under the Romans!" ever entered popular discourse?
I'm aware that the monophysites quite welcomed Muslim's conquest in 7th century and there were lots of cooperation between Muslims and Christians in Middle East, especially in Abbasid's rule.
In short, yes. But I don't know if this generally pre-dates the emergence of modern nationalism. It's not rare today for Maronite Lebanese Christians, for instance, to insist that they are "Phoenician" rather than "Arab". Copts in Egypt similarly sometimes emphasize their Arabness, sometimes emphasize that they are the true descendants of the Ancient Egyptians (their language, Coptic, is directly descended from the language of the Ancient Egyptians). Assyrian Christians often emphasize their association with Assyria (as well as, to what is as far as I can tell a less degree, to the Syriac speaking Christians of the Roman Empire). Chaldean Christians (also Syriac speaking) often emphasize their connections to Babylon rather than Assyria. Greeks in the Ottoman Empire were known as Rum since they were the descendants of the (Eastern) Roman Empire. Their early nationalism focused as much on Constantinople as Athens, if not even more so. The Megali Idea of the Greek Nationalists was in part to revive the glory of Byzantium.
Here's a little bit on the history of the Assyrians (and by extension, the Chaldeans, though that's more discussed in the comments). Here's more on the Megali Idea.
Again, I don't know if there has been a continuous knowledge that "things were so much better when we ruled!", but there have been consistent naming practices of these groups throughout Muslim rule (Rum Romans, Assouri Assyrians) that there must have been some collective attachment to historical greatness, at least among the elites (especially interesting is the case of the Chaldeans, who only really emerged out of a split within the Assyrian Church and named themselves in the 16th/17th centuries, which is before the onset of nationalism in the region, something that really only dates to the 19th century). I am confident that all of this attachment to pre-Muslim greatness only increases in the late 19th and early 20th centuries. It was not limited to non-Muslim groups, however. The Turkish government in the first half of the twentieth century pressed (largely ahistorical) connections to the Hittites and Sumerians. Two of the largest state development banks formed in this period, for example, are Etibank and Sümerbank. This was done in part to counter Greek and Armenian claims to Anatolia. Anatolia was an essential part of Greece since pre-Roman times, and of course the center of the Byzantine Empire; Armenian history similarly is more centered around Lake Van and Mt. Ararat in Turkey than Yerevan in modern Armenia, which was something of a backwater.
tl;dr: I can't tell you if they said things like "Things were better under the Romans" (well, I'm sure that's in plenty of Ottoman Greek writings, but I don't know if it's true generally), but there was plenty of awareness that "We are intimately related to these pre-Muslim groups", especially after the rise of nationalism in the 19th century.