For the sake of parsing the sentence, we can temporarily remove the phrase "as a Squib" as that is not integral to the structure of the sentence. So instead we have:
who | could no more | have Stunned the fireworks | than swallowed them
The word "who" is introducing some fact about Filch, namely the fact that will be mentioned in the rest of the sentence. That fact relates to two things that he is unable to do. The phrase "could no more" sets up the relation between the two things that Filch cannot do. Had the sentence merely been trying to convey that Filch could not do two things, it could simply say:
who could not have Stunned the fireworks, nor swallowed them
who could not have stunned or swallowed the fireworks
However, the sentence is actually trying to convey a measurement of what Filch cannot do. The sentence takes for granted that it is obvious that a man could not swallow fireworks. Knowing this, the sentence then tries to illustrate that it should also have been obvious that Filch could not stun the fireworks. To this end the sentence says that he could not have done one thing any more than he could have done the other thing. I.e. in the same way that he obviously cannot swallow the fireworks, he can also obviously not stun them.
If we wanted to rephrase this in a slightly easier way, we could perhaps write:
who could not have Stunned the fireworks any more than he could have swallowed them
Or perhaps even easier (but less elegant):
who could not have stunned the fireworks just like he could not have swallowed them
now that we understand the sentence structure, we can reinstate the phrase "as a Squib". This phrase doesn't add anything to the necessary sentence structure; it is simply the explanation of why the rest of the sentence is true. As you know from the earlier books, a Squib is someone who cannot do magic. Thus, it is saying that the reason why Filch could obviously not Stun the fireworks just like he could obviously not swallow them is that Stunning is magic and Filch is incapable of magic.
Knowing this we can gain a greater appreciation of the entire passage. The passage is subtly pointing out aspects of both Umbridge's character and Filch's character. They both know that Umbridge is the one who attempted to stun the fireworks, and they both know that it is not even conceivable that Filch could have been the one to Stun them. Yet despite this, Umbridge is still haughty enough to blame Filch, and Filch is meek enough to accept the blame, even though they both know that it was actually Umbridge's fault. This now gives us new insight into their character traits, and particularly their relationship with each other.