Likely yes, but not for long
Firstly, it is possible to master the Elder Wand through accident or trap:
A full century later, another unpleasant character, this time named Godelot, advanced the study of Dark Magic by writing a collection of dangerous spells with the help of a wand he described in his notebook as "my moste wicked and subtle friend, with bodie of Ellhorn, who knowes ways of magick moste evile".
Godelot is known to have perished in his own cellar, where was locked by his mad son, Hereward. We must assume that Hereward took his father's wand, or the latter would have been able to escape, but what Hereward did with the wand after that we cannot be sure.
The Tales of Beedle the Bard, pages 101 and 103
Secondly, Muggles can use a wand... in a very limited manner:
[...] Muggles cannot perform magic. The best - or worst - they could hope for are random and uncontrollable effects generated by a genuine magical wand, which, as an instrument through which magic is supposed to be channelled, sometimes holds residual power that it may discharge at odd moments
The Tales of Beedle the Bard, page 82
Thirdly, the Elder Wand is extremely powerful and has a lot of Dark Magic knowledge:
Believers in the Elder Wand, however, hold that because of the way in which it has always passed allegiance between owners - the next master overcoming the first, usually by killing him - the Elder Wand has never been destroyed or buried, but has survived to accumulate wisdom, strength and power far beyond the ordinary.
The Tales of Beedle the Bard, pages 102 and 103
Fourthly, the wand's power only remains as long as it has a master:
“Aren’t you listening? Snape never beat Dumbledore! Dumbledore’s death was planned between them! Dumbledore intended to die undefeated, the wand’s last true master! If all had gone as planned, the wand’s power would have died with him, because it had never been won from him!”
The Deathly Hallows
Now, this is entering the realm of conjecture, but:
The Muggle that killed the previous owner would become the new master of the Elder Wand. While Richard's comment infers that the wand runs a quick "who's the better wizard" test, I believe this is incorrect, as Harry wasn't a particularly powerful or skilled wizard, and Voldemort was extremely powerful... but the Elder Wand chose to stick with the less-powerful owner rather than transferring allegiance to the more powerful wizard.
But, the first time the Muggle tried to use it, there would be an extremely powerful backfire of magic that would likely kill the Muggle. (Ron's spello-taped wand completely obliviated Lockhart due to a backfire... imagine what would happen with the Elder Wand!)
The wand would then be in a state of being its own master, with no allegiance to any owner, which would signify the end of it's power.
Also, AFAIK, there is no canon reference to the Elder Wand only passing to a wizard: Dumbledore's notes to The Tale of the Three Brothers in The Tales of Beedle the Bard only mentions the words "owners", "masters" and "man" when referring to ownership of or transferring ownership of the Elder Wand:
Believers [...] hold that because of the way in which it has always passed allegiance between owners - the next master overcoming the first
The Tales of Beedle the Bard, page 102 (emphasis added)
What must strike any intelligent witch or wizard [...] is that every man who claims to have owned it
The Tales of Beedle the Bard, page 104 (emphasis added)
There is no canon reference to only a wizard being able to own a wand/the Elder Wand.
It is also possible for the Elder Wand to transfer ownership through an accident or trap.
∴, it is possible to gain ownership of the Elder Wand.
However magic 'backfires' when a Muggle uses a wand, due to 'residual magic' within the wand.
The Elder Wand has had centuries of storing 'residual magic'.
∴, the backfire would likely kill the Muggle.
The Elder Wand would then be its own master, and it's power would be broken.