TL;DR: Pretty unlikely.
The objects are small (and hence faint), there aren't very many of them (astronomically speaking) and they're moving very fast so there isn't much time to spot them. Even if you knew they were coming, it might be tricky to catch a glimpse of them with today's technology. Basically, it would depend more on luck than judgement.
If a whole bunch hit Earth, we'd be more likely to notice, but the odds are even slimmer.
Lets consider visible-light astronomy. This isn't totally unreasonable; the sail probably isn't very warm (being small, shiny and fast moving).
The faintest near-earth object in this JPL database is 2008 TS26, with an absolute magnitude of 33.2. We can find the absolute magnitude of our solar sail by approximating it as a lambertian disk with geometric albedo $a$ of 1 (this is wrong, but doing it right is quite a lot harder, so it'll do for now) and a diameter in kilometres $D$ of 0.004:
$$H = 5\log_{10}\left({1326 \over D\sqrt{a}}\right)$$
This gets us an absolute magnitude of about 27.6... about a hundred times brighter.
2008TS26 has a semimajor axis of 1.92AU. Assuming that it is in opposition to the sun (a syzygy, an awesome word that is hard to use very often) it will have an apparent magnitude of 34.4, given that $$m = H + 5\log_{10}\left({D_{BS}D_{BO} \over D_0^2}\right) - 2.5\log_{10}\left(q(\alpha)\right)$$ where $H$ is the absolute magnitude, $D_{BS}$ is the distance from the body to the sun, $D_{BO}$ is the distance from the body to the observer, $D_0$ is the distance between Earth and the Sun and $q(\alpha)$ is something called the phase integral that I'm declaring to be 1 in this position.
With the same geometric relationship, we can rearrange the equation to find the equivalent distance of our solar sail where it would have the same apparent magnitude:
$$10^\frac{m - H + 2.5\log_{10}(q(\alpha))}{5} = D_{BS}^2 - D_{BS}$$
Leaving us with a nice quadratic to solve, giving us a $D_{BS}$ of ~5.35AU, the point at which we can start to see it with something capable of spotting 2008 TS26. We can solve a second quadratic with the non-constant parts $D_{BS}^2 + D_{BS}$ to find when we would theoretically stop seeing it, if the sun were somehow transparent, which would be 4.35AU on the far side of the sun.
These represent fairly optimistic figures, I think. They're slightly implausible, representing a trajectory that transects the Earth and the Sun, but you can perhaps imagine a closely grazing path which would behin and end at pretty similar distances, I think. An incoming probe travelling at 0.2c will cross that 9.7AU distance in approximately 41 minutes and 35 seconds. The points of peak visibility will be at the beginning and end of that traversal, and at some point in the middle when it is edge-on to observers it will be basically invisible. Any other trajectory through the solar system will have closer start and end of visibility points (because sunlight won't be maximally reflected towards us) and so the observation time will be reduced, but where the probe, Sun and Earth are at right-angles you might get a much brighter and easier to spot object. Too many variables, really. You might consider asking this question in Astronomy.SE where people might have a better idea of things like our observation capabilities.
Realistically, that gives us a few hours in which to have a suitably powerful telescope pointed in precisely the right direction (as such a powerful telescope will have a comparatively small field of view). I'm not sure how many such scopes exist, but the odds seem pretty vanishingly small, to be honest.
if success only depends on a very narrow margin, the probes might be slightly larger (not by orders of magnitude!) or their speed might slightly differ from the 0.2 c in the question.
I'm not sure what the minimum flyby time would be to guarantee detection by current Earthbound observers, but I strongly suspect that it would be so long as to require very, very slow probes, and the chances of a civilisation bothering to fire such things out and hope they still work and people still care about them in a thousand years seems slim.
Would this change if the arrival of such probes was theorized/expected, and special equipment was made beforehand to scan for them?
Maybe. The duration of the sail constellation flyby will be short, the objects will be hard to spot, and they'll likely to be widely separated. This seems like a pretty fearsomely complex challenge even with this huge advantage, given the potential margins for error.
If there is no other way to detect them, would this change if one (or a few) of them actually hit the planet?
The sails pack a pretty substantial punch... about 4.3kT TNT equivalent. I don't think you'd get gamma rays, but you'll definitely get an interesting bang. I'm not sure what would be about to detect such a thing, but high-atmospheric flashes seem more likely to be spotted than distant, dim, tiny objects. If observed, the nature of the bang would probably be suspicious. I'm not sure of our capability to spot small objects entering the atmosphere, and this object certainly wouldn't leave a contrail so would be harder to spot (but more interesting if it were).
Regarding Ghedipunk's comment:
The lasers used to accelerate the craft may be visible in the target system.
The laser array is powerful, at 100GW, but it only operates for 10 minutes per sail. That's a few hours for the whole constellation in which to spot the launch. The beams will be focussed on a spot very close to Earth (certainly within an AU) and so will be so enormously diffuse by the time it reached the target system.
I'm not quite sure how diffuse though. It is possible that some alien exoplanet survey might spot something funky happening with Sol's brightness during that very brief window of opportunity.
