No, a squib (or a muggle) cannot brew a potion.
As revealed in the latest update on Pottermore, although potion-making doesn't appear to have a great deal of magical input, the reality is that at some point in the process wandwork is essential to add magic to the ingredients.
"It is often asked whether a Muggle could create a magic potion, given
a Potions book and the right ingredients. The answer, unfortunately,
is no. There is always some element of wandwork necessary to make a
potion (merely adding dead flies and asphodel to a pot hanging over a
fire will give you nothing but nasty-tasting, not to mention
Since a squib is incapable of performing wanded magic, they would also, by definition be incapable of brewing a working magical potion.
This was backed up in an interview in 2006, albeit one largely focused on muggles rather than squibs:
Q: Can muggles brew potions if they follow the exact instructions and they have all of the ingredients?
JKR: Well, I'd have to say no, because there is always...there are magical component in the potion, not just the ingredients. So, at some point they will have to use a wand. I've been asked what would happen if a Muggle picked up a magic wand in my world, and the answer would probably be something accidental... possibly quite violent. Because wands, in my world, is merely a vehicle, a vessel for what lies inside the person. There is a very close relationship -- as you know -- between the wand that each wizard uses and themselves. In fact, we'll find out more about that in book 7.
For a muggle you need the ability, in other words, to make these things work properly but you're right and I think that's an interesting point. Potions seems, on the face of it, to be the most Muggle-friendly subject. But there does come a point in which you need do more than stir.
The question still remains whether a squib or a muggle could assemble previously magicked potion components and combine them into a working potion but the overall impression seems to be no, they could not.