You are absolutely correct, this would happen with satellites. In fact, it happens to a degree on Earth, especially if the sunset is redder than normal due to, say, a volcanic eruption. But first, let me just point out a few facts.
- A Lunar Eclipse is 1000 times (Or more) dimmer than at peak. Furthermore, there is a much larger difference than lit by the sun.
- Any satellite by necessity is traveling very fast.
- The period of time this would happen corresponds to the speed it is traveling. Depending on the direction of orbit, and assuming a typical half hour long sunset/sunrise on Earth, the window for a red looking spacecraft would be about a minute per orbit. The actual time when there is primarily red light is much less, only a few minutes on the ground or a few seconds for the orbital period..
So, if someone were to deliberately set about to take such a picture, they would need to find the exact right time, and somehow take the picture during a very brief period of time where the spacecraft would look red. Also, it would largely look black, as the amount of light would decrease rapidly, making the timing even more difficult. So, in short, it is possible, but would require very careful timing, a high gain camera positioned to get it just right, and probably some trial and error. But, with 90 minutes between sunsets in orbit, it could be done. I just haven't heard of a case of it actually being done.