In the Harry Potter universe, do these beings have souls in the same sense as wizards:
Quotes from books or canonical sources such as Pottermore are preferred.
Inspired by this question on Dementors - my answer there depends on an assumption that Muggles also have a soul. My own research did not give any definite proof
Muggles and Squibs can be assumed to have souls.
For evidence, let's consider Lupin's description of what happens to those who lose their souls to the Dementors:
'You can exist without your soul, you know, as long as your brain and heart are still working. But you'll have no sense of self any more, no memory, no ... anything. There's no chance at all of recovery. You'll just - exist. As an empty shell.'
Harry Potter and the Prisoner of Azkaban - p.183 - Bloomsbury - Chapter 12, The Patronus
This then is what a wizard's soul is. Their sense of self. Them. Their memories even, which apparently are lost with the soul, even if the brain is still working. Since Muggles have senses of self and memories, and souls provide these to wizards, it certainly seems like Muggles would have them too. That seems to be what the soul is and does in the Harry Potter universe.
Also, consider this exchange in The Order of the Phoenix.
'But what ARE Dementoids?' asked Uncle Vernon furiously. 'What do they DO?'
'I told you - they suck all the happiness out of you,' said Harry, 'and if they get the chance, they kiss you -'
'Kiss you?' said Uncle Vernon, his eyes popping slightly. 'Kiss you?'
'It's what they call it when they suck the soul out of your mouth.'
Aunt Petunia uttered a soft scream.
'His soul? They didn't take - he's still got his -'
She seized Dudley by the shoulders and shook him, as though testing to see whether she could hear his soul rattling around inside him.
'Of course they didn't get his soul, you'd know if they had,' said Harry, exasperated.
Harry Potter and the Order of the Phoenix - p.36 - Bloomsbury - Chapter 2, A Peck of Owls
Of course, the story is told from Harry's point of view and it could be that Harry simply didn't realise that Muggles don't have souls and hence this exchange can happen. But, it seems an odd scene for the author to write and never again comment on or correct if she intended to create a universe in which only wizards have souls.
Finally, consider how death itself works in the Harry Potter universe, with reference to how Horcruxes work:
'Well, you split your soul, you see,' said Slughorn, 'and hide part of it in an object outside the body. Then, even if one's body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged.'
Harry Potter and the Half-Blood Prince - pp.464-5 - Bloomsbury - Chapter 23, Horcruxes
Death, then, is when the soul no longer has an earthly form to inhabit and must pass on.
Now, of course, it's unclear whether Harry Potter animals have souls and they certainly live and die, so maybe Muggle death is different. We don't know for sure and wizards do live longer, but they are also human and can be born to Muggles. We also see that Avada Kedavra works on Muggles too, despite not damaging the body in any way.
The police had never read an odder report. A team of doctors had examined the bodies, and had concluded that none of the Riddles had been poisoned, stabbed, shot, strangled, suffocated or (as far as they could tell) harmed at all. In fact, the report continued, in a tone of unmistakable bewilderment, the Riddles all appeared to be in perfect health - apart from the fact that they were all dead. The doctors did note (as though determined to find something wrong with the bodies) that each of the Riddles had a look of terror upon his or her face - but as the frustrated police said, whoever heard of three people being frightened to death?
Harry Potter and the Goblet of Fire - p.9 - Bloomsbury - Chapter 1, The Riddle House
Then again, Avada Kedavra also works on a fox, which is an animal, and we see animals such as birds being created with charms and animals are killed and eaten and put into potions without any problem, which doesn't sit too happily with the idea of them having souls.
However, on preponderance of evidence, I'm gonna say Muggles and Squibs have souls. In particular, I consider Lupin's description of a wizard without a soul to indicate that the soul is one's personality - perhaps even one's humanity.
Also, if we jump out of universe, JK Rowling stated in Harry Potter and Me:
Death is an extremely important theme throughout all seven books. I would say possibly the most important theme.
She is writing a book which she considers to be about death. That's what she's talking about. It seems only logical that we the audience are supposed to take it that all these ideas about death being but the next great adventure are supposed to have some relevance to us the readers. It's simply too much for me to imagine that Muggles and Squibs in her world don't have souls.
Inferi do not have souls, surely.
They are just corpses. They are just puppets. Consider how Snape contrasts them with ghosts:
'The Inferius is a corpse that has been reanimated by a Dark wizard's spells. It is not alive, it is merely used like a puppet to do the wizard's bidding. A ghost, as I trust that you are all aware by now, is the imprint of a departed soul left upon the earth ... and of course, as Potter so wisely tells us, transparent.'
Harry Potter and the Half-Blood Prince - p.431 - Bloomsbury - Chapter 21, The Unknowable Room
We have just seen that death is the removal of the soul from the body. And so certainly the dead body would not have had a soul when it was reanimated, and it sure seems like ensouling is a bit beyond these spells. I say that because a Horcrux keeps a soul earthbound and so prevents death. This requires serious and rare dark Magic, which rather suggests that there aren't many spare souls floating around looking for bodies. And to create a soul ex nihilo, or pull one from the world beyond, would surely be to create life, rather than a puppet.
Why would't they? Harry sees Dudley's soul almost sucked out by the Dementor in Order of the Phoenix. If Muggles have souls, so do Squibs - if you're basing it on not having Magic. Squibs can see Hogwarts after all.
And, what does Voldemort do with Dementors if not let them feast on Muggles?
There are several facts that support Muggles having souls:
Several of Voldemort's Horcruxes were created with the murder of Muggles. There doesn't appear to be any distinction between Muggles and Wizards/Witches for whatever magic is involved. If a Horcrux creation requires a human soul be separated from its body, a reasonable interpretation of the murder requirement, then Muggles must have souls.
Frank Bryce, a Muggle, was one of the people (or whatever they were at that point) who came out of Voldemort's wand during Prior Incantatem.
It is stated that killing Muggles does not create ghosts. However, this is because a Wizard's death releases magic energy that can create an "imprint", and Muggles do not have the magical energy to create an imprint of their soul, not because they don't have a soul.
A Beast is one of the three classifications used by the Ministry of Magic to catalogue the various magical creatures that inhabit the wizarding world. Loosely defined, a Beast is a magical creature that does not have sufficient intelligence to understand the laws of the magical community nor bear part of the responsibility in shaping those laws ... There are even extremists who campaign for Muggles to be classified as Beasts
http://harrypotter.wikia.com/wiki/Beast
The fact that people who classify Muggles as "beasts" are considered "extremists" suggests that the mainstream position is that Muggles are considered Being (albeit nonmagical Being). (There being three divisions, into Beast (have bodies), Spirit (have souls), and Being (have both)).
Also, my understanding is that Squibs are socially distinct from Muggles (e.g. the Statute of Secrecy doesn't prohibit doing magic in front of them), but they are, as far as their essential nature, the same as Muggles.
Inferi are merely animated corpses, and are described as being "puppets". Puppets don't have souls.
