There is little account and evidence of Squibs in the world.
Merope Gaunt: (not actually a Squib)
In Harry Potter and the Half-Blood Prince, we see in one of Dumbledore's Pensieve memories (The House of Gaunt) that Merope Gaunt owned a wand.
She was not a Squib, but her father (Marvolo Gaunt) thought that she was; I think that in this case, it is fair to assume that if Squibs weren't allowed to (or couldn't) own a wand, Marvolo would not bother with a wand for her.
This leaves us with a couple of points to consider though:
- If Squibs weren't allowed wands; Did Marvolo know, or care?
The Gaunts were fiercely segregated (by choice or otherwise) from the general and the Magical communities; if there was a law or rule or anything similar to stop Squibs from having or owning a wand, then they may not have known about it, hence why Merope had one.
- Did Merope know she wasn't a Squib?
Merope may have known all along that she wasn't a Squib, and therefore may have found herself a wand to use.
Angus Buchanan: (wrote My Life as a Squib)
...As the time approached for him to leave to attend Hogwarts School of Witchcraft and Wizardry, his sister Flora forged an acceptance letter and Angus, frightened of his father, played along, going as far as going to Diagon Alley and buying a wand and pretending it had chosen him ... 1 [emphasis added is mine]
At this point, Angus has bought a wand; meaning that they can obviously possess one; but the wand has seemingly not chosen him.
Filch clearly cannot do magic. As demonstrated multiple times:
... “Don’t Stun them, Filch!” shouted Umbridge angrily, for all the
world as though it had been his suggestion.
“Right you are, Headmistress!” wheezed Filch, who was a Squib
and could no more have Stunned the fireworks than swallowed them.
-Harry Potter and the Order of the Phoenix (Chapter Twenty-Eight - Grawp)
...Umbridge and Filch had been observed trying different
means of removing it but without success. Eventually the area was
roped off and Filch, gnashing his teeth furiously, was given the task of
punting students across it to their classrooms. Harry was certain that teachers like McGonagall or Flitwick could have removed the swamp
in an instant, but just as in the case of Fred and George’s Wildfire
Whiz-Bangs, they seemed to prefer to watch Umbridge struggle. ...
-Harry Potter and the Order of the Phoenix (Chapter Thirty - Grawp)
Does he own a wand? Not sure...
Keeping up Appearances
It's just strange that Umbridge tells Filch not to Stun them (the fireworks). She likely didn't know that He (Filch) was a Squib. But it brings about my next thought: A Squib likely would want to make it appear that they are not a Squib by purchasing and brandishing a wand. [No proof or citations for this one.] So he may have attempted magic with a wand but clearly failed. Is this because he is a Squib with no magical abilities, or the wand hadn't chosen him so he could not perform magic. See next thought.
Now, assuming that you mean "mastering" when you say 'own' a wand:
As answered here, (although the question is aimed at wands being won), the same rule applies: a wand 'generally' and 'usually' bends its will to its new master, but doesn't always have to!
I believe that a Squib could hold a wand as Angus Buchanan did, and not quite be able to own it.
1 Harry Potter Wikia - Angus Buchanan