As <PERSON> pointed out we learned in "Harry Potter and the Half-Blood Prince" that living creatures can be turned into Horcruxes.
The Argument against it was:
A phoenix could probably be a Horcrux. However, given that the Horcrux would be rendered useless when the phoenix regenerated, and that a phoenix wouldn't be likely to want to be a Horcrux anyway, making a phoenix into a Horcrux would be a poor choice.
At this point, I think we have to ask ourselves the question if a reborn Phoenix is really dead/destroyed beyond magical repair since this is the only possibility to destroy a Horcrux.
The Horcrux-receptacle has to be destroyed BEYOND REPAIR [...]
— JKR on Twitter: https://twitter.com/jk_rowling/status/563633021837131777
I would argue, that - since a Phoenix is a creature which is really hard to domesticate - it's body is replaced but its soul and memories are still the same after being reborn. Otherwise, that would mean that e.g. <PERSON> would have to domesticate <PERSON> each time he is reborn (which sounds to me too much work, even for <PERSON>).
The phoenix gains an XXXX rating not because it is aggressive, but because very few wizards have ever succeeded in domesticating it.
— Fantastic Beasts and Where to Find Them
I would argue that this shows that a phoenix is not damaged "beyond repair".
So here I think it depends. <PERSON> is a snake and therefore has to molt from time to time. One could argue that molting is a similar process as being reborn in terms of a Phoenix. That would mean, that when being killed/destroyed by e.g. the killing curse, the Horcrux inside the Phoenix would survive the process.
On the other hand, hitting a living creature which is a Horcrux with a killing curse does not necessarily kill the creature. An example of that is <PERSON> himself, who was hit by a killing curse but survived while the curse killed the Horcrux inside him.
Also <PERSON> argues often, that the Horcrux-Receptacle has to be destroyed beyond repair which in case of living creatures is the body of this creature.
<PERSON>: So, can I ask this? This is kind of a random question but if <PERSON> had this Horcrux in him, of course, sort of, would he have actually have died, like say when a dragon could've killed him, or when he was falling during Quidditch, or anything?
<PERSON>: Well, you've got to-- if his body had been irreperably destroyed, he has to die to get rid of that piece of soul. His body has got to be irreperably damaged. 773
So a lot of people asked, and I think I've answered this since... but a lot of people immediately said, having finished "Hallows", "(gasps) But then, that means, in Chamber of Secrets when he was pierced by the basilisk..." But no, no, no, no. He didn't die! He didn't die! That was stated right at the beginning with the Horcrux. The receptacle has got to be destroyed. His body wasn't destroyed. He got a bit poisoned, and then he got the antidote immediately. So, you know, that's not gonna drive out this piece of soul.
— Interview: http://www.accio-quote.org/articles/2007/1217-pottercast-anelli.html
So while the body of a molted snake is not damaged beyond repair, the body of a reborn phoenix is! That for me is the Argument, that turning a Phoenix into a Horcrux is a very bad idea.
On the (last) other side. <PERSON> body was still perfectly fine after <PERSON> hit him with the killing curse, so according to this theory the Horcrux would have theoretically survived. It didn't so... ;) 773