According to this scene, Vision can lift Mjolnir irrespective of being worthy due to being a machine. Since the Eternals are also ultimately automatons in the MCU, would they be able to lift Mjolnir as well?
To me, the scene in Avengers: Age of Ultron where the Vision lifts Mjolnir implies that he did so because he was worthy, not because he was an android. At the time, he was trying to convince the Avengers that he was on their side (rather than Ultron's), but they weren't sure if they could trust him or not. Then, just as he picked up Mjolnir, he said...
VISION: So there may be no way to make you trust me, but we need to go.
Avengers: Age of Ultron (2015)
... and the Avengers immediately stopped questioning whose side he was on, as though the very sight of him lifting Mjolnir was enough to make them trust him. That only makes sense if they thought he lifted Mjolnir because he was worthy, rather than him being able to bypass the worthiness enchantment due to being an android.
This interpretation is supported by what the director and writer, Joss Whedon, had to say about the scene in a podcast interview with Empire:
"That came from a 'Do you know what would be cool?' moment. It’s the cheer moment of the film. And what’s great is that, like the Hawkeye thing, we’d set it up – we’d unknowingly set it up, just by having that ['Who is worthy enough to Mjolnir?'] sequence, then with Quicksilver as well, trying to grab it and it throws him off. Both of those things were in the script before I came up with the idea of [the Vision lifting Mjolnir].
I had done something similar in an episode of Angel, where I needed you to know someone was telling the truth. It was, very simply, Angel saying, 'He hates it if you ask questions, he can’t lie.' So, you just accept that.
So on the one hand, I want them all to trust each other and go into battle not as a coherent group, but when they finally all show up at the church, they really do come together for the first time. On the other hand, I need them to take this guy with them, and I need something to say, 'All right, we’re off!' And that really does answer a lot of questions. It was so much fun and so cute. Chris added the 'Nice work!' as he walked by Tony, by the way.
With that in mind, I don't think the Vision lifting Mjolnir proves anything about whether any of the Eternals could or not. Nor is there presently any other evidence I'm aware of that proves anything in that regard, one way or the other.
Also, it's not entirely clear that the Eternals genuinely are synthetic beings. That's what they were led to believe by Arishem, but in the first post-credits scene in Eternals, Starfox is introduced as both an Eternal and Thanos' brother.
PIP: Behold, the royal prince of Titan, brother of Thanos, the Knave of Hearts, defeater of Black Robert...
PIP: Oh. Defeater of Black Roger. The great adventurer, Starfox of Mystery Planet.
STARFOX: What a pleasure to make your acquaintance, my fellow Eternals.
Judging by the evidence presented in this answer, the MCU version of Thanos isn't a synthetic being, but one who was naturally born from biological parents. So, if Starfox really is his brother -- and note that Thanos is pictured with his parents and one sibling in concept art supplied within the linked answer -- he presumably isn't a synthetic being either.
As there's conflicting information on this point, we can't be sure of the truth either way, until more evidence comes in. That said, it wouldn't exactly come as any great surprise if it transpired that Arishem lied to the Eternals about their origins.
In same part of the Marvel Universe in the comics, certain very VERY powerful beings could lift Mjolnir, but in the MCU it's been made very clear that it holds to the classic idea that its enchantment prevents it being lifted but nobody save whom is deemed "worthy".
The only possible way around that is that someone would have to be more powerful than Odin, who created the enchantment. So the question then would be "Are the Eternals more powerful than Odin?" and I'm prepared to bet the answer is no.
Now if you want to discuss the Celestials, possibly Galactus, or the higher beings in the MCU like the Living Tribunal or even Eternity themself, we might be able to go with a "maybe".