In Harry Potter and the Chamber of Secrets, Harry and Ron find out that the Hogwarts caretaker, Argus Filch, is a Squib.

"And what on earth's a Squib?" said Harry.

To his surprise, Ron stifled a snigger.

"Well – it's not funny really – but as it's Filch ..." he said. "A Squib is someone who was born into a wizarding family but hasn't got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual. If Filch's trying to learn magic from a Kwikspell course, I reckon he must be a Squib. It would explain a lot. Like why he hates students so much." Ron gave a satisfied smile. "He's bitter."
_{— Harry Potter and the Chamber of Secrets by J.K. Rowling, Chapter 9, "The Writing on the Wall"}

Filch reluctantly talks about it in front of a number of teachers, such as Snape and Dumbledore.

"[...] He found—in my office—he knows I'm a—I'm a—" Filch's face worked horribly. "He knows I'm a Squib!" he finished.

"I never touched Mrs. Norris!" Harry said loudly, uncomfortably aware of everyone looking at him, including all the Lockharts on the walls. "And I don't even know what a Squib is."

"Rubbish!" snarled Filch. "He saw my Kwikspell letter!"
_{— Harry Potter and the Chamber of Secrets by J.K. Rowling, Chapter 9, "The Writing on the Wall"}

Yet Umbridge did not seem to know he was one.

"Don't stun them, Filch!" shouted Umbridge angrily, for all the world as though it had been his incantation.

"Right you are, Headmistress!" wheezed Filch, who as a Squib could no more have Stunned the fireworks than swallowed them.
_{— Harry Potter and the Order of the Phoenix by J.K. Rowling, Chapter 28, "Snape's Worst Memory"}

Umbridge had a Squib brother and a Muggle mother, with whom she broke all ties, so it doesn't seem likely she would be considerate enough to cover for a Squib — it's likely she just didn't know.

So who all knew that Filch was a Squib?

TL;DR: various teachers, and at least some students, but not (originally) Diary-Riddle.

At least some students

Word of the attacks spread among the students after Colin was attacked.

The news that Colin Creevey had been attacked and was now lying as though dead in the hospital wing had spread through the entire school by Monday morning. The air was suddenly thick with rumor and suspicion. The first years were now moving around the castle in tight knit groups, as though scared they would be attacked if they ventured forth alone.

_{Harry Potter and the Chamber of Secrets, Chapter 11, "The dueling club"}

Although the exact basis on who's going to be attacked are clear ("Mudbloods"), everyone is afraid, save maybe for Slytherins; Neville especially, even if he's a Pureblood, thinks his lack of magical skill might make him a target.

Meanwhile, hidden from the teachers, a roaring trade in talismans, amulets, and other protective devices was sweeping the school. Neville Longbottom bought a large, evil smelling green onion, a pointed purple crystal, and a rotting newt tail before the other Gryffindor boys pointed out that he was in no danger; he was a pureblood, and therefore unlikely to be attacked.

'They went for Filch first,' Neville said, his round face fearful, 'and everyone knows I'm almost a Squib.'

_{Harry Potter and the Chamber of Secrets, Chapter 11, "The dueling club"}

No offense, but second-year Neville isn't the brightest kid around. It's likely the info was leaked, or deduced by someone else and spread as a (on-point, for once) rumor.

Hagrid

It's revealed later on in the series that Hagrid knew, too, but he wasn't in the mentioned scene in the question. It doesn't indicate, though, when he learned the info.

"And what difference does that make?" asked Filch obnoxiously.
"I'm a ruddy teacher, aren' I, yeh sneakin' Squib!" said Hagrid, firing up at once.

_{Harry Potter and the Half-Blood Prince, Chapter 19, "Elf Tails"}

For what it's worth, the individuals present when Filch told about the Kwikspell letter were Harry, Hermione, Ron, Lockhart, Dumbledore, McGonagall and Snape.

Math time: not (originally) Diary-Riddle

This one speculates a bit, but I think it holds. It's been pointed out in the comments that Ginny, or at least Diary-Riddle, knew that Filch was a Squib:

‘Ginny Weasley opened the Chamber of Secrets. She strangled the school roosters and daubed threatening messages on the walls. She set the serpent of Slytherin on four Mudbloods, and the Squib’s cat.’

_{Harry Potter and the Chamber of Secrets, Chapter 17, "The Heir of Slytherin"}

Whether Mrs Norris was an actual target or a happy accident is debated. To be honest, I think both answers make very good points. One might argue that young Riddle indeed knew that Filch was a Squib, then let's do something one shall never do when dealing with Harry Potter: math.

Peeves has been in Hogwarts for a long time, as pointed out by both answers to Why was Peeves in Hogwarts? By long, I mean that he was probably already there when Filch took the job (confirmed - his Pottermore page mentions him being there in 1876). In the Battle of Hogwarts, there is this passage:

“P-Peeves?” stammered Filch as though he had never heard the name before.

“Yes, Peeves, you fool, Peeves! Haven’t you been complaining about him for a quarter of a century? Go and fetch him, at once!”

Filch evidently thought Professor McGonagall had taken leave of her senses, but hobbled away, hunch-shouldered, muttering under his breath.

_{Harry Potter and the Deathly Hallows, Chapter 30, "The sacking of Severus Snape"}

Filch has then been working at the school for likely 25 years (even if there's room for exaggeration from McGonagall). Filch/Peeves' relationship doesn't evolve much during the series, so I think it's safe to say that Peeves was here before Filch, and Filch complained from day one of the job, which would then be circa 1970.

Voldemort was born in 1926 (per Pottermore), so his Hogwarts years would be 1937-1944 (give or take one year). That's 30 years before (alleged) Filch's hiring.

Further evidence was provided by Alex in chat:

Mrs. Weasley grinned, her eyes twinkling. “Your father and I had been for a nighttime stroll,” she said. “He got caught by Apollyon Pringle — he was the caretaker in those days — your father’s still got the marks.”

_{Harry Potter and the Goblet of Fire, Chapter 31, "The third task"}

More math! Arthur Weasley is born in 1950, so Pringle would've been caretaker in the 60's-70's, before Filch - Voldemort is even more unlikely to have known Filch.

My conclusion is that Diary-Riddle knew that Filch was a Squib from Ginny, via the rumors that spread among her fellow students.

Last but not least: don't underestimate the rumors

Talking about rumors... Based on real-life teenager behavior, it's likely that rumors of Filch being a Squib existed waaaay before the Chamber of Secrets events. Filch is a bully in a school of teenagers. He regularly makes them clean up stuff without magic, openly loves the good old ways of dungeons, chains and other fun.

Teenagers are prone to spread rumors and talk ill of people they don't like - Filch qualifies, and even if Squibs aren't that much known (Hermione didn't seem to know!), every school has these "Mate, I'm telling you, Filch is deffo a Squib" sayings going.

I reckon even the dumbest student could not assume Filch was a proper wizard. He's never seen to perform any magic and it seems he doesn't even have a wand. So there are only two possibilities: either he is a legit muggle or a squib. I assume being a squib is more shameful - that's why he doesn't want anyone to know for sure he is a squib, but still one do not need additional information to at least assume he is. So there should have been rumors about that in every intake. I think being a squib is more of his personal tragedy rather then some actual secret for teachers and students.