I did toy with this idea before when one of my x-ray tetras was fatally ill but ultimately didn't proceed. In my own case, the fish concerned had catastrophic swimbladder malfunction which was causing him to bounce and catapult all round the tank and, in that unique/unusual situation, I thought being confined in a small container might be distressing for him. If the sick fish weren't moving much, or had a swimbladder malfunction that might have it lying on its side / upside down but otherwise relatively still, then this would be an ideal solution.
As for new fish: If it's new fry, I have read of some breeder boxes having holes that are quite large and the fry end up escaping through that and ultimately get consumed by others. If it's new fish from the LFS, then you wouldn't need this / they may become distressed being in a small container and it wouldn't serve the purposes that a quarantine tank would (ie keeping them apart in separate water to ensure they're disease-free and don't pass on any illnesses to existing tank stock).
In a nutshell, breeding boxes can be very useful in some circumstances, but not all.
For sick fish, I have a plastic tank with a spare heater and filter and silk plants specifically for it - it's stowed away at present but is something like 20L. For anyone reading this and contemplating buying one for such purposes, I would recommend getting one from a LFS such as PAH or MA, not from a homestore - I once bought one from "The Range" but it wasn't practical because the lid couldn't accommodate cable wires, an issue I hadn't foreseen.