To even contemplate firing Boone or Cashman is ludicrous and a knee-jerk reaction.
Firing Boone would be not a kneejerk reaction, he should not have been hired to begin with. He is adequate, he is OK, he is the guy who sits in the next cubicle to you for a few years who you are never impressed by but one day he is inexplicably promoted over you and then later you learn his dad and granddad worked at the company for decades and are friends with many people in upper management. His biggest strength by far is not screaming at reporters when asked idiotic questions in press conferences, that doesn't exactly help boost the winning percentage. He has had a ton of talent to work with (Fangraphs thought NY had the most talented MLB roster coming into this season, and they have been in the top few MLB teams every year under Boone) and he has won a lot of regular season games but he has never been a good decision maker under pressure, when quickly changing circumstances of a game force him to improvise a bit from the elaborate game plan he is presumably supplied with before each game. This quality is less than ideal for a guy whose most important job is postseason decision maker. He has pretty clearly lost the clubhouse as Terry has talked about, IMO he has never really been an asset but adequate, a guy you hope you can win despite of.
Cashman is more complicated, he has been there so long that I'm not sure how NY would even go about replacing him. His tenure has been up and down, but at least at this point, it's pretty easy to see that he had a terrible offseason (imagine if he had gotten even one of Schwarber or Gallo or Lance Lynn, for starters).
On the flip side, and part of this is because there are fewer minor league teams per franchise now, but holy shit is the farm loaded, with another top international stud supposedly signing in January (SS Roderick Arias). NY already has 5 other SSs in their system who deserve serious top 100 consideration based on pedigree/performance (Peraza, Volpe, Vargas already on at least one list, Park and Smith both have OPSs over 1.000 currently). Cashman has to get credit for that too, and with every other AL East team also having quite a bit of talent on the way, I'm sure Cashman is hesitant to cut into his own stockpile in fear of it backfiring on him in a few years.
But Cashman sticking by Boone is one of the worst moves he's made IMO, I hope he realizes that sooner rather than later. My theory (posted here for the first time) is that unless Boone miraculously saves this season, he will be fired after NY is done and Cashman will hire Joe Espada, who was with NY from I think 2014-2017 (special assistant to Cashman and then 3rd base coach) and then was hired by the Astros as bench coach to replace Cora where he still is. Clearly with Houston playing great, he cannot move midseason but that is my guess as to why Cashman is dealing with this situation the way that he is. Of course he could still fire Boone now (hurrah) and put in an interim guy (please not Nevin, please not Nevin, please not Nevin) but that would just complicate things in the offseason so if Espada is his plan, I kind of get not firing Boone until the season is done.