It would appear that illithids likely do have souls, and therefore can be captured in a Ring of Mind Shielding
Illithids themselves at least don't believe that they have souls
In Volo's Guide to Monsters, p. 80, under the Divine Magic subsection of the Mind Flayer chapter, it says the following:
Illithids acknowledge the existence of divine entities, but it is unusual for any but a deviant mind flayer to actively worship such a power. Since the are capable of planar travel, illithids don't view the afterlife and the Outer Planes in the mythic way that most other races do. Illithids don't believe they possess souls whose eternal fate is governed by the gods. Instead, when a mind flayer's brain is returned to the elder brain to be consumed, the creature's intelligence lives on. Only if an illithid's brain isn't retrieved after death would its consciousnesses be cast into oblivion.
It's clear that illithids themselves don't believe they have a soul given the above quote. On the other hand, it mentions that their "consciousness [would] be cast into oblivion"; what exactly that means I am unsure about, since that might imply that, regardless of what they believe, they do end up in some kind of afterlife, which would in turn imply that they do have souls, or this might just be a poetic way of saying that they are dead.
Slight counterargument: Illithids just don't believe in divine judgement
An alternative reading to the above could be that illithids actually do believe they have souls, they just don't believe that their souls will be judged by the gods. Consider this reading:
Illithids don't believe they possess souls whose eternal fate is governed by the gods.
In other words, they believe they possess souls, but not souls whose "eternal fate is governed by the gods". In either case, this is just a reading on what mind flayers think is true, and isn't actually hard evidence on what is actually true.
That said, it is possible for a mind flayer to achieve lichdom, which requires them to have a soul
On pages 171-172 of VGtM, it describes alhoons and illithiliches, both of which are described as having phylacteries.
From p. 171 about alhoons:
Dreadful Deliverance. Lichdom offers salvation and the prospect of being able to pursue knowledge indefinitely. Having feasted on the brains of people when alive, a mind flayer has no compunction about feeding souls to a phylactery.
From p. 172 about illithiliches:
The path to true lichdom is something only the most powerful mind flayer mages can pursue, since it requires the ability to craft a phylactery and cast the imprisonment spell.
If we couple this with information on liches from elsewhere, such as in the Monster Manual (p. 203):
A lich is created by an arcane ritual that traps the wizard's soul within a phylactery.
then we must conclude that illithids, despite what they typically believe, do in fact have souls, else they would not be able to achieve lichdom.
Therefore, if they have a soul, then it is possible for the soul of a mind flayer to be trapped within a ring of mind shielding.
Slight counterargument: the alhoon description avoids the term "soul"
I have, however, noticed that not once in the description of alhoons or illithiliches does it actually explicitly mention that the mind flayer has a soul. Instead, it refers to their mind, implying that they might have a form of consciousness that can even exist beyond death but that is separate from a soul:
Precarious Immortality. Unlike with true lichdom, the periapt of mind trapping doesn't restore the alhoons to undeath if they are destroyed. Instead, a destroyed alhoon's mind is transferred to the periapt where it remains in communication with any other trapped alhoon minds, as well as the souls of those sacrificed.
It mentions the souls of the sacrifices plenty in these descriptions, but never the soul of the mind flayer. I'm not sure if that's significant, but it could be thought of as a flaw to my above conclusion. Then again, this could simply be a detail pertaining to how an alhoon is different from a "true" lich, an illithilich, of which there is precious little information to be found in this book.
