It seems that there are two distinct stages of damaging the soul. The mere act of murder is said to rip apart the soul. This is what Slughorn tells Tom Riddle in Chapter Twenty Three of Half-Blood Prince:
"By an act of evil — the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: He would encase the torn portion —"
And this is what Dumbledore indicates to Snape in Chapter Thirty Three of Deathly Hallows:
“If you don’t mind dying,” said Snape roughly, “why not let Draco do it?”
“That boy’s soul is not yet so damaged,” said Dumbledore. “I would not have it ripped apart on my account.”
Muggles can presumably achieve the state of having a ripped apart soul, as they can commit murder just like wizards can. However, Dumbledore's statement in Chapter Thirty Five of Deathly Hallows about Harry being a "Horcrux" is very specific:
“You were the seventh Horcrux, Harry, the Horcrux he never meant to make. He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child. But what escaped from that room was even less than he knew. He left more than his body behind. He left part of himself latched to you, the would-be victim who had survived.
This implies that Voldemort had gone beyond the mere ripping apart that happens to anyone who murders. By creating previous Horcruxes, Voldemort had rendered his soul entirely unstable.
Hermione's statement in Chapter Six of Deathly Hallows also implies that it is specifically the act of creating Horcruxes that makes the soul so unstable:
“And the more I’ve read about them,” said Hermione, “the more horrible they seem, and the less I can believe that he actually made six. It warns in this book how unstable you make the rest of your soul by ripping it, and that’s just by making one Horcrux!”
If this is true, then Muggles should not be able to qualify for the inadvertent "Horcrux" situation. This is because even though they can rip apart their souls by murdering, they can't render their souls entirely unstable by making other Horcruxes (as that requires a spell).
Additionally, the piece of Voldemort's soul seems to have only latched on to Harry because Voldemort couldn't die (because of his other Horcruxes). But if someone who could die was in the same situation, they would just die. If the mere fact that they murdered would be enough to keep them alive then all murderers would be immortal (at least until whoever possesses their soul dies), something which we do not find in any of the books.
Or perhaps put differently: When a regular murderer (with no prior Horcruxes) gets killed he simply dies. Even though his soul has been ripped apart because of the murder, there is nothing tethering his soul to this world, so his soul would simply do what anyone's soul does when they die. However, if someone had a prior Horcrux then when he gets killed he won't die, because the Horcrux is tethering his soul to this world. In such a case there could be the potential for a piece of his soul to break off and latch on to another living being. However, it still seems that this would not happen to someone who only had one Horcrux. It seems from Dumbledore's statement that Voldemort's soul was especially unstable because of the multiple Horcruxes.
One could also argue that there is no fundamental difference between a soul split simply from murdering, and a split soul that was turned into Horcruxes. The reason then that Voldemort's soul was so unstable would be the multiple murders rather than the Horcruxes themselves.
In any case, this type of situation shouldn't be able to happen to a Muggle because a Muggle wouldn't have a prior Horcrux to tether his soul to this world. Even if his soul was rendered unstable it would not latch on to another being.