is there any justification that make watching the sky on these occurrences (equinox/solstice) better than any other day?
Not really. The winter solstice has the advantage that it is a nice long night (assuming you're not observing from the equator, or your planet has a negligible axial tilt, etc) but that's about it.
Would it be believable that it can only work on these days because it is when another celestial body out there is out of the way and so not blocking its view?
It would be rather odd that something should move out of the way precisely on those four days, and then immediately move back. Obviously if the something is technological, that some other explanation can be made for its behavior, but it doesn't sound natural.
Imagine that this mechanism has to locate a very specific point in the sky.
One thing, visible from four widely spaced points on the planet's orbit, but not visible at any other time? I'm not really buying it, but I suspect I'm a lot more nitpicky than most, and have a harder time suspending my disbelief.
So what's to be done?
One solution that allows for nearly arbitrary amounts of contrived situations is to have something technological, that does some specific odd things at specific strange times for whatever reason. This may or may not fit your theme, and it does also shift the problem from "why observe at these times" to "why do a thing at these times", so it doesn't really fix the underlying problem.
You can handwave in some other process that occurs four times a year that just so happens to coincide with the solstices and equinoxes. The period of a conjunction with an inner planet, possibly. I don't know if having the timing be a coincidence is something you'd find particularly satisfying, though.
If you were happy to have it fire once a year, then it would be reasonable to have something with a slightly odd structure, say a deeply buries observatory of some kind, with a viewing shaft that only points at The Thing at a specific time each year, and during the rest of the year it just isn't visible. Such a system might have been built to last over a very long period of time (perhaps like the clock of the long now), hence its slightly unusual structure. It can't have been built to last for too long though, because axial precession is a thing and over the timescale of millenia the apparently fixed stars will move... either your target will not be visible through the device at all, or it will appear on a day other than the solstice, etc.