I'm reasonably sure everything used in Hogwarts is produced in the magical world, possibly using (you're not going to believe it) magic. As such, I don't think wizards can produce ballpoint pens, because that would require them to follow some advancements in technology, which they famously don't do. Remember Mr. Weasley, and how everyone thinks he's a wacko.
My point is - the supplies in magical world (including stationery) must be produced magically due to wizards' alienation from Muggles; I'm not saying a pen wouldn't work, I'm simply saying wizards wouldn't think of using one. Or chalk it up to quill-makers' lobby.
As to why Muggle-borns don't use them - what if they do? Since the series is centered very narrowly on the trio, there's not much room for others' perspective. Maybe some Muggle-borns do use them, we just don't see it (observer bias). There's also the problem of anti-Muggle sentiment, which seems to exist even in late 90s when the series is set. I assume showcasing one's Muggle-bornness would not be a very wise choice, especially with Voldemort's growing influence in later books.
The issue of wizarding pride is mentioned in the Pottermore article on technology, written by J. K. Rowling:
There is another reason for most wizards' avoidance of Muggle devices, and that is cultural. The magical community prides itself on the fact that it does not need the many (admittedly ingenious) devices that Muggles have created to enable them to do what can be so easily done by magic. To fill one's house with tumble dryers and telephones would be seen as an admission of magical inadequacy.
I'm guessing a quill and an ink bottle are also easier to use in conjunction with magic - one can enchant the quill to write, regularly dipping itself in ink, and maybe sharpen itself. How does one refill a ballpoint pen? I don't. I throw it away, and get a new one; or if I really like it, I get a new core and change them. But the core of a pen is the part that is actually the hardest to make, what with the small ball, tiny tube, and the ink. The schematics of a quill are just simpler.
And why stop on pens? Why not use A4 paper, like normal people do, instead of parchment (and why not measure the parchment in metres, like normal people do, instead of feet)? I'm guessing the scope of possible improvements is too wide, and people just don't care.
One could also point out that glasses, watches, and some other tools are used by wizards. I'm guessing that is due to them being invented before the International Statute of Secrecy, which was accepted in 1692.
The Hogwarts Express is a notable Muggle invention that is being used by Wizards. However, this one is very much an edge case. As it is explained in its Pottermore article written by J. K. Rowling, the train was the best option possible when it came to transporting a lot of students at one time, while still being hidden from Muggles. Multiple solutions, including various flying transportation methods, portkeys, and sometimes Apparition were tried, and some were meant to be tried but were blocked by the Headmasters of Hogwarts as breaching the security of the castle. In the end, it took a Muggle-curious Minister of Magic, "one hundred and sixty-seven Memory Charms and the largest ever mass Concealment Charm performed in Britain" to steal the train and put it to use. It was by no means easy, and definitely met opposition at the time:
The Hogwarts Express underwent several magical modifications before the Ministry approved it for school use. Many pure-blood families were outraged at the idea of their children using Muggle transport, which they claimed was unsafe, insanitary and demeaning; however, as the Ministry decreed that students either rode the train or did not attend school, the objections were swiftly silenced.
There's also the problem of very few Muggle-borns, who would know about pens, being in charge of things. Sure, some students are Muggle-borns, but how many of the Ministry officials, or even the Hogwarts staff, are also Muggle-born? I would assume a Muggle-born taking over the ministry could change things for the best, but that would require a sequel to the seven Harry Potter books, and we haven't gotten one.
There is also a relevant Quora thread, which says more or less the same things, and maybe more.