How should I take the concept of 'non-entity'?

An entity has independent, separate, or self-contained existence. Emptiness, in its various interpretations in the differing schools of Buddhism, refers to the absence of those qualities. So emptiness could be defined as non-entity-ship of all appearances. In the materialism/physicalism prevalent in European/Western philosophy, entity-ship is a foundational understanding.

Yep, I think you got it right, in this passage Hume identifies being a self with being the subject of perception (or we can say, experience). So, to him, having no perception equates to not existing, even if temporarily (during sleep).

In Buddhism, we don't define it the same way. Identifying perception as the self, or really anything impermanent as self, is a source of dukkha.

Now, the question of entities plays a prominent role in Mahayana Buddhism where all entities (not just self) are understood to be transient bundles of causes and conditions, without any intrinsic essence at their core, hence Emptiness (Shunyata).

I'm not very familiar with Hume but it seems like he is saying something similar about the self being a bundle? In both cases there's no need to become a perfect non-entity because being a bundle means there was no entity to begin with.

I have not come across this concept of a “perfect non-entity” in Theravada. From a pure logical perspective, it does not make sense. First, a “perfect non-entity” only makes sense in the context of an “imperfect entity” (similarly for concepts like existence/non-existence, perception/non-perception and ego/non-ego). This means that this concept is dependent on an imperfect entity or else it will be meaningless. If something (even a concept) is dependent on another for its existence, then it cannot be perfect.

Second, the reason why an entity is imperfect in Buddhism is because it is dependent on other causes/conditions for its existence. Inherently, it does not have an intrinsic and independent nature. So, this means a perfect non-entity must have an intrinsic and independent nature. In the context of Buddhism, everything in samsara is dependently originated. Even non-living phenomena like the seasons, the winds and rain are dependently originated, not to mention living beings like you and me. Therefore, the question is can there be a perfect non-entity in samsara? In Buddhism, my understanding is only temporarily but not permanently. An example is a living Arahant whose existence in samsara ends upon Parinirvana. If something perfect exists in samsara permanently and interacts with it then the entire definition of samsara is dubious.

Lastly, I don’t think we can ever grok a concept which is beyond our experience, no matter how hard we try to imagine or extrapolate it. An analogy would be someone who lived all their life floating on the sea, they may see the shore (a perception) but they would never understand what it is like to be dry and standing on solid ground (an experience). Another aspect of independent and intrinsic state such as the Deathless state is that they are orthogonal to known samsaric states. For example, the Deathless state may contain characteristics opposite of the samsaric states but it would also contain characteristics that has no opposites and vice versa. Because samsara and Nirvana did not existed for one another, they are independent.

Even the concept of Emptiness (Sunyata) is different when experienced and perceived by an Arahant versus as experienced and perceived by a non-enlightened being. While I believe that the philosopher Hume is alluding to something that is independent and intrinsic in nature; unless he is enlightened and talking about the Deathless element, the closest Buddhist states to his theory would be the sphere of Nothingness or the realm of asaññasatta where no perception exists. However, as Buddhists we understood neither of these attainments are perfect. With Metta.

I believe this 'non-entity' concept is laid out in the Maalunkyaputta Sutta

Well then, Maalunkyaputta, in things seen, heard, sensed, cognized: in the seen there will only be the seen, in the heard only the heard, in the sensed only the sensed, in the cognized only the cognized... Then, Maalunkyaputta, there will be no 'thereby' for you. Having no 'thereby' you have no 'there.' Having no 'there,' Maalunkyaputta, there is for you neither this world, nor the next, nor anywhere in between. That in itself is the end of suffering.

By forbearing to form thoughts on things seen, heard, sensed, cognised, no self-consciousness is generated and one is self-emptied as in suññatā.

An entity is a form. Therefore, to consider "non-entity", we find that the Buddha says:

AN9.41:11.2: ‘Why don’t I, going totally beyond perceptions of form, with the ending of perceptions of impingement, not focusing on perceptions of diversity, aware that “space is infinite”, enter and remain in the dimension of infinite space?’

Beyond that, the Buddha introduces:

AN9.41:12.2: ‘Why don’t I, going totally beyond the dimension of infinite space, aware that “consciousness is infinite”, enter and remain in the dimension of infinite consciousness?’

Beyond that, the Buddha introduces:

AN9.41:13.2: ‘Why don’t I, going totally beyond the dimension of infinite consciousness, aware that “there is nothing at all”, enter and remain in the dimension of nothingness?’

Beyond that, the Buddha introduces:

AN9.41:14.2: ‘Why don’t I, going totally beyond the dimension of nothingness, enter and remain in the dimension of neither perception nor non-perception?’

And finally, the Buddha steps away from attachment to any of these dimensions, saying:

AN9.41:16.1: As long as I hadn’t entered into and withdrawn from these nine progressive meditative attainments in both forward and reverse order, I didn’t announce my supreme perfect awakening in this world with its gods, Māras, and Brahmās, this population with its ascetics and brahmins, its gods and humans.

In other words, the Buddha did not get stuck in the search for the perfect non-entity. He let it go.