They would certainly not be admitted, and probably wouldn’t enjoy their experience if they did attend.
Admittance to Hogwarts is controlled by the Quill of Acceptance and the Book of Admittance. You name gets written in the book if you demonstrate signs of magical ability while you’re young, but this has to be your own magic: it’s not enough to ride off the coattails of your parents.
This is part of the sample content on Pottermore, and here are the relative parts:
These are the Quill of Acceptance and the Book of Admittance and they constitute the only process by which students are selected for Hogwarts School of Witchcraft and Wizardry. […] The Book and Quill's decision is final and no child has ever been admitted whose name has not first been inscribed on the book's yellowing pages. […]
[The Book’s] track record in keeping Squibs out of Hogwarts is perfect. Non-magic children born to witches and wizards occasionally have some small, residual aura of magic about them due to their parents, but once their parents magic has worn off them it becomes clear that they will never have the ability to perform spells.
If you can’t go to Hogwarts if your name doesn’t appear in this Book, and your name will never be written in this Book if you’re a Squib, then you can’t go to Hogwarts. End of discussion.
But would a Squib want to go to Hogwarts? Would it be beneficial or healthy for them to do so? I suspect not; they’d always be an outsider. This snippet of conversation is fairly telling:
“Squibs were usually shipped off to Muggle schools and encouraged to integrate into the Muggle community… much kinder than trying to find them a place in the Wizarding world, where they must always be second class, but naturally [redacted] wouldn’t have dreamed of letting her daughter go to a Muggle school—“
— Deathly Hallows, chapter 8 (The Wedding)
For examples, look at how Filch is treated by students, or Mrs. Figg at the Ministry hearing. The magical society doesn’t rate Squibs, and they’d be social outcasts at Hogwarts. Consider how certain students treat Muggle-borns, and now imagine how they’d treat Squibs.
You’d be a tiny minority, in a school of people with special talents that you don’t share, and the entire experience would remind you of what you were missing out on. Large aspects of their school experience are cut off to you, and there are plenty of ways to be bullied that you have no way to retaliate against.
And you go through all this, and what comes out at the end? There are very few jobs in the magical community that you can take, so you’ll probably end up in the Muggle world. A brief glimpse of what you could have had, but never will. It seems cruel.
I suspect a Squib at Hogwarts would be miserable and bullied, and would leave school with a strong inferiority complex. Even if they want to attend, I think it’s better to turn them away.
The link in DVK’s answer to J.K. Rowling’s old website says something similar: Squibs at Hogwarts would never be able to really fit in.
Squibs would not be able to attend Hogwarts as students. They are often doomed to a rather sad kind of half-life (yes, you should be feeling sorry for Filch), as their parentage often means that they will be exposed to, if not immersed in, the wizarding community, but can never truly join it.