You certainly can break some monitors by inputting a frequency outside of what they can handle. Whether a monitor is sensitive to this and what happens depends on the design of the monitor. Digital monitors will almost all shut down. Usually, if something breaks, the HOT just dies, but sometimes other stuff breaks, too.
The bootup screens on a PC are not 24k, they are 31k (VGA). Handily, this is exactly double what a standard res arcade monitor is designed for (15-15.5k). Just about every monitor I've found designed for that range will happily lock it's PLL onto every other sync pulse giving the classic "double screen" effect. If your monitor is reliably doing this, you should generally be fine, but some monitors will apparently dutifully try to run all the way up at 31k, usually with the result being a blown HOT.
Dumping 24k into a 15k monitor can be a problem, though. What happens will depend on the monitor, but sometimes you can nudge the horizontal operating frequency outside the range of what the actual output stage can deal with, and things break. I wouldn't say it's overly common, but I've certainly done it (by accident, and to an old fixed frequency workstation monitor, not an arcade monitor). It was easy enough to fix, at least.
I don't know if the JPAC actually does an every-other-pulse output or not. It's often not necessary. The monitor will do that for you.
FWIW, there's less inherent "protection" on most monitors designs against weird vertical frequencies. Incidentally, 240p at ~90Hz works out to ~24kHz, but if you actually try feeding that into a medium res arcade monitor, you probably won't like the results (but then again, you may be fine). Note that many PCs run their bootup screens at ~72Hz. The line count is dropped to ~400 visible lines to keep the horizontal scanrate at ~31kHz. This will probably make some monitors unhappy.