Satellites are somewhat closer to the Moon than the Earth is (Or rather, can be), and might well have the sun blocked from them. Furthermore, they can move somewhat more than the Earth, and thus have the potential for more eclipses. Are solar eclipse a significant concern for the design of satellites (Specifically, the Moon blocking the Sun)?
Orbiting satellites have to deal with darkness all the time when Earth is between the satellite and the sun, and these periods last for much longer than an eclipse does for the satellite. So although I'm no expert, I expect that any satellite designed to function above the dark side of the planet would have no problems with an eclipse.
During non-eclipse times it is normal for satellites to experience "solar outage", where the signal is cut or degraded, but not for more than a few minutes a day. This is caused by the Sun lining up directly behind the satellite (from Earth's perspective).The solar energy interferes with the signal/our ability to "read" that signal. In an eclipse, the source of that interference (the Sun) would actually be blocked by the Moon). If there is any kind of outage caused by the eclipse it would also only last a few minutes (the length of the eclipse time). So, likely, if the eclipse causes any loss/degradation of signal, it would no worse than during the normal solar outages, if there's any effect at all. And the solar outages really are not a big deal.
Satellites commonly go through the shadow of the Earth, and thus are able to deal with those outages. The question of being in the Moon's shadow really is how can that affect the ability to recover from being in the Earth's shadow. The question really is, how long will such an eclipse affect the satellite? Let's look at a few orbits, and compare the Earth/ Moon distances. I'm going to use the max width of a lunar eclipse, 267 km, as a reference. I'm also going to double that, as the penumbra will have effects.
LEO (800 km)- 100 minute orbit, The Earth eclipse time max is around 34 minutes. Eclipse max is less than 2 minutes. Probably not a significant effect.
GEO- 72 minutes eclipse time, at least 3 minutes of eclipse time possible. Could be a bit higher if at the closest point towards the Moon, as the altitude will make a big difference, but only another minute or two.
Bottom line, unless a satellite is already at margin, and has an absolute worst case experience in an eclipse, it will not have a signficant effect.
The Gaia mission is in an orbit around the Earth-Sun L2 point. That orbit had to be very carefully designed to avoid ever passing through the Earth's shadow because that would cause an unacceptable temperature change in the satellite (which needs to be in an extremely stable environment to do its job). I think it does pass through the Moon's shadow a few times but that has a much smaller effect (because the Moon is smaller) which they can allow for.
darkness does not affect geostationary satellites, however they have been affected by lack of solar energy coming from sun since they have a large amount of equipment attached to the satellite for functioning which cannot rely on batteries for such long period of time during eclipse.