It is common knowledge that Death Eaters and Voldemort alike hate Muggles, Muggle-borns and Squibs, so is there a canon reason why Filch is still employed during Deathly Hallows?
The main target for the prosecutions during Deathly Hallows are Muggle-born wizards.
From Harry Potter and the Deathly Hallows, chapter 11 (emphasis mine):
“Meanwhile, the Ministry has started moving against Muggle-borns.”
Lupin pointed at the Daily Prophet.
“Look at page two.”
Hermione turned the pages with much the same expression of distaste she had worn when handling Secrets of the Darkest Art.
“‘Muggle-born Register,’” she read aloud. “‘The Ministry of Magic is undertaking a survey of so-called “Muggle-borns,” the better to understand how they came to possess magical secrets.
“‘Recent research undertaken by the Department of Mysteries reveals that magic can only be passed from person to person when Wizards reproduce. Where no proven Wizarding ancestry exists, therefore, the so-called Muggle-born is likely to have obtained magical power by theft or force.
“‘The Ministry is determined to root out such usurpers of magical power, and to this end has issued an invitation to every so-called Muggle-born to present themselves for interview by the newly appointed Muggle-born Registration Commission.’”
“People won’t let this happen,” said Ron.
“It is happening, Ron,” said Lupin. “Muggle-borns are being rounded up as we speak.”
The main charge against Muggle-borns is that they have somehow "stolen" magic.
Squibs are exactly the opposite - they are pure-bloods who have somehow "lost" their magic. This does not apply to them. Squibs could also be used for the cause of rounding-up Muggle-borns by being presented as the "victums" - the pure bloods who lost their magic because some evil Muggle "stole" it.
On top of this - Filch is harmless and insignificant in general. There is little point in rounding him up.