As noted in the Puzzle 24 Solution Page, each of the columns of puzzle 24 consists of one or two words which are hidden in the headlines of other puzzles. So first, solve all the other puzzles. In the list below, answers follow their corresponding hidden words; answers in square brackets are confirmed by solving puzzles, and answers between asterisks are answers written on the copy of puzzle 24 I originally received. Answers in braces are answers written on puzzle 24 for puzzles which we do not have the information to solve, so I am taking them for granted.
Each of these answers is the same length as its corresponding hidden word, so they are meant to replace those hidden words in the dropquote.
Now, solve the dropquote. Our best attempt is as follows (possibly LAST/PATH and TREK/OPEN should be swapped).
I think these are instructions for modifying a set of runaround instructions we don't have. Mark Gottlieb's thesis says that the endgame of this Hunt "was made up of two puzzles masquerading as normal puzzles (one in the regular Hunt and one in the impossible section) and hints hidden, among other places, in the jokey answer page to the impossible section." This entire "impossible section" is missing.
Each double black square is the end of a sentence. The first sentence says to start in building 14, not in building 1. This probably puts them in front of Hayden. If they start here, there is only one sensible turn (turning left and going outside would be a turn before the far wall). This takes them into building 2. The next two sentences probably make more sense given the instructions, as does "run out" in the following sentence, but four times they will "run out" and take some number they see there. These get ordered as indicated, and the most frequent digit is removed every time it appears. What's left, we know from the Hunt map, is the number of an Athena workstation in 2-225 which probably is not there any longer.