The compression joke evokes a misconception (as STEM jokes often do) which, in this case, is that breaking a hash means finding a way to recover the full text of a multi-megabyte document, from a 100-something byte hash.
Whereas all it means is that it's become feasible for someone to produce a fake document with the same hash as the genuine one; the attack depends on the existence of collisions, which depend on the hash being one-way, such that an infinite set of documents could be "recovered" from the hash.
Most of the documents you could "decompress" from a hash are going to be gibberish, but some of them are going to be legit. If a hash is "broken", that means it's somehow unduly easy to search that ocean of gibberish collisions to find a chosen text which goes to a certain desired hash.
Nothing quantum can redefine a function that isn't one-to-one and onto into one that is. The hashing function remains ever the mathematical object that it is: a one-way function. But calculations of or searches through that function can be faster.
In a post-quantum world, pre-quantum crypto and hashing could well turn out to be "for shit" as such, but not due to ruses like suddenly reversing irreversible functions.
Not saying that the joke isn't funny! We can imagine some hard-core computer science or math sitcom in which the fall characters say things like this, and only the very narrow target audience gets it.