edit: see qed1 and WelfonaShelf (heh) below for better discussion. I'm a bit more out of my area than I initially thought on this.

I can talk about the manuscript history of Euripides. I will leave the education level of the cinema audience in the 1960s to someone else.

Kirchoff is the O.G. daddy for Euripides manuscript tradition. He was really the first to do a thorough modern academic investigation, and he found the following: nine Euripides plays had been selected by the early Byzantines as "canon," which included Andromache, Hecuba, Hippolytus, and Medea. Bacchae and Iphigenia at Tauris were not on the list. From this Byzantine "canon," it might easily have happened that the rest of the plays got lost and disappeared. Instead, Kirchoff theorizes that a single manuscript from the 9th or 10th century CE (now lost) contained no less than 19 of the original plays, a Christmas miracle. Note that the 9th century CE is just about as far back as we can trace most manuscripts of ancient writings. This original wondrous book was copied around 1100 CE, also now lost, but with only the nine "Byzantine canon" plays. This copy went on to be the best-kept archetype for most of our Euripides plays that we have today. There are three main manuscripts, a Marcianus in St Mark's in Venice, a Vaticanus (in the Vatican), and a third in Paris. Medea was present in all of these, so yes, in the 12th century, Euripides Medea at least existed and was in some sort of circulation among European elite. It seems probable that Eleanor and Richard would have been able to get a copy, if they wished. It is certainly in the realm of the possible, if not probable. I don't myself know if Eleanor would have read Greek. She certainly read and wrote Latin, and we have several famous letters to and from her written in Latin. We would need a Medievalist to talk about elite proficiency with ancient Greek in the 12th century.

The other 10 plays that we still have today come to us via a separate line of succession, again from this original lost 9th-10th century archetype. There are two manuscripts, both from the early Renaissance.

The discussion about the manuscript tradition of Euripides in Greek by u/Alkibiades415 and u/qed1 here is certainly fun (who doesn't love medieval transmission of ancient literature??). But I think Euripides’ play is a bit of a red herring here. The question should rather be, would Henry and Eleanor know about Medea, even if they couldn’t read Greek and had never heard of Euripides? In that case the answer is clearly yes.

Medea was a popular character in ancient literature, far beyond the play by Euripides. In Latin, her story takes up most of Book 7 of Ovid’s Metamorphoses, and she’s the subject of a play by Seneca. Both Ovid and Seneca were still well-known in the medieval Latin west. I’m not sure about the medieval transmission of Seneca’s plays, but his philosophy and letters were a standard part of Latin education (and part of modern Latin education too! Argh, flashbacks of translating his letters…)

As for Ovid, complete manuscripts of the Metamorphoses are known from the 11th century, and there are numerous earlier fragments. From the 12th century up to the time of the first printed editions in the 15th century, there are over four hundred hand-written manuscripts. I don’t know if we can point to any single manuscript and say aha, this is the one that Henry or Eleanor would have read, but it is definitely possible that they would have known the story of Medea from the Metamorphoses.

Even more likely though is that they would have known it from French literature, of which Eleanor was a major patron. For example, the Roman de Troie, a history of the Trojan War, was written with Eleanor’s patronage by Benoît de Sainte-Maure in the 12th century. Benoît based his poem on two Latin texts that were attributed to Dares Phyrgius and Dictys Cretensis (who were supposedly ancient Greek historians, but really they were medieval inventions...we had to translate those guys in school too, haha), as well as on book 7 of the Metamorphoses. Although it’s about Troy, the framing device of the Roman is, in fact, the story of Jason and Medea! So it seems beyond question that Eleanor would have been familiar with it.

Eleanor and Henry probably had no idea there was an ancient Greek playwright named Euripides, or that he had written a play about Medea, and they would never have seen a performance of it. Presumably nothing like the conversation in The Lion in Winter ever really happened. But Eleanor, at least, would have known the story from French and Latin literature.

Sources:

Ruth Morse, The Medieval Medea (Boydell & Brewer, 1996)

Ovid's Metamorphoses, ed. William S. Anderson (University of Oklahoma Press, 1998)

Tamara F. O’Callaghan, “Tempering scandal: Eleanor of Aquitaine and Benoît de Sainte-Maure’s Roman de Troie” in Eleanor of Aquitaine: Lord and Lady, ed. Bonnie Wheeler and John Carmi Parsons (Palgrave Macmillan, 2003)

Would an educated 12th century individual be familiar with Euripides?

In short, no. Prior to the renaissance^tm there was no real interest in Greek literary works in the Latin world. They might have been familiar with Euripides as an author, if only because Jerome was and makes a handful of references to him through his works. For example, in his commentary on Daniel, he references the Medea with relation to a textual issue in Daniel 6:4:

Furthermore Euripides in his "Medea" equates the word amplakiai ["offenses"] (spelling it with a p instead of a b) to hamartiai, that is to say, "sins."

Likewise, in his discussion of the visions, he suggests that Sophocles and Euripides flourished at around the time of the Persian wars.

These references were, of course, still available to Latin authors, and from a quick search of some databases, most of the twelfth century references that I can find to Euripides in Latin can be pretty obviously tied to Jerome. (Although there are references to Euripides also in Boethius, Calcidius, Cassiodorus, Lactantius and a couple other widely read latin texts.)

As I say, Latin authors of the period were not especially interested in Greek literary figures, so it is not especially common to find references even to this sort of thing. (By comparison, references to Homer or ancient Greek philosophers (well besides Plato and Aristotle who were actually be read in translation), particularly the Seven Sages, are commonplace.) But just to illustrate the point we can look at a reference in Peter of Blois and John of Salisbury, both of whom were well integrated at one point or another into the English court: Peter was a corresponded of Eleanor and wrote a number of letters on her behalf and, on the other side of the fence, John was secretary to Thomas Becket.

Both make reference to Jerome's comment in his tract Against Jovinianus that: "All the tragedies of Euripides are a curse upon wives" (totae Euripidis tragcediae, in mulieres maledicta sint) in their respective (probably not so-) casually misogynistic polemics against marriage. (Peter's is found in letter 79 and John's in Policraticus 8.11, which was incidentally excepted and translated into the vernacular as a standalone text in subsequent centuries.)

But to speak to the core of the question, no knowledge of Euripides was required to make such a reference to Medea. She was a well known classical figure, and her story would be totally familiar to anyone who had a decent familiarity with Latin literature. Her story can be found at least twice in Ovid (Metamorphoses 7 and Heroides 12) as well as in a wide range of casual references, for example in Virgil's eighth Eclogue.

So, for example, Medea is one of the examples that Alan of Lille, another contemporary of Matilda, cites of woman "inverting the rules of Venus" in The Plaint of Nature 8.10:

Medea, a stepmother to her own son, destroyed a wonderful little product of love that she might accomplish a terrible work of love.

Similarly, in his rather antisemitic reading of Song of Songs 1:5, Philip of Harveng (d. 1183) compares the Jews to Medea as a mother who has abandoned her motherhood through hatred of her children:

And who of sound mind understands this rightly about Judea, who is the mother and loves not, just as the stories hold about Medea: a mother for whom there was no love of her children, no care, but inflamed by hate she submitted them to untimely death.

Et quis sani capitis sane sentiat de Judaea, quae mater est, nec diligit sicut fabulae perhibent de Medea; cui matri filiorum nulla fuit dilectio, nulla cura, sed accensa odio illos de medio morte sustulit immatura (PL 203, 233)

So that sort of reference is doesn't depend on any familiarity whatsoever with Euripides.

Edit: I see that /u/WelfOnTheShelf has simultaneously produced something far clearer and more coherent than my ramblings here. I'll leave this up though in case anyone is interested in the random anecdotes I've pulled together here.