If your GPS receiver had an onboard atomic clock (imagine a \$50,600 cell phone, \$600 for the iPhone and \$50,000 for the atomic clock), could you do away with the fourth satellite with the atomic clock on board the receiver?
Yes, in theory. Actually in normal GPS, four signals will normally not give a unique solution, even with zero error bars, but usually only one solution will be close enough to the surface of the earth to make sense.
It seems that chip-scale atomic clocks (CSAC) are now down to about \$1500. However, their precision is not good enough for them to work for long periods of time without recalibration, and that recalibration would normally require that you get four GPS satellites again. The spec sheet for a current model says "<9E-10 /mo Aging Rate (Typical)," which suggests that if you ran the thing for a month without reacquiring signals from four satellites, you would be off by the end of the month by a few milliseconds, which is an error of hundreds of kilometers.
To go one step further, if you assume you are in the surface of the earth I think you might be able to get away with only TWO satellites (although in this case there are two solutions on the surface of the earth that are both reasonable and choosing the correct one is harder). This wouldn't work for a plane and would require a very precise knowledge of the Earth's topography.
I don't think this works very well. The intersection of three spheres is typically two points. With normal GPS, you get multiple solutions with 4 satellites, but you can usually tell which one is real because it's near the earth's surface. With only two satellites plus the CSAC, you would get two solutions with no obvious way to tell which was correct.
The problem with a GPS satellite is it literally tells an adversary where it is at any time making it inherently easy to attack and destroy. An advantage with the on-board atomic time piece approach is one could work with a minimal GPS constellation [...] you could determine a receivers position with only three entities revealing their position as opposed to the conventional four.
I don't really see how this helps. A GPS satellite that doesn't reveal its position isn't functioning as a GPS satellite, so it might as well have been put out of commission. Also, the most likely method that's been discussed for destroying GPS satellites is as a result of a nuclear explosion in space. You can get EMP or the trapping of high-energy charged particles in the earth's magnetic field. The latter has already happened unintentionally in 1962 with the Starfish Prime nuclear test. In this type of scenario, an attacker doesn't need to know the satellite's position to high precision -- they may not need to know its position at all.
Maybe a more compelling application is that you don't want to lose GPS tracking if you're in a deep canyon where you only have a limited view of the sky. This happens to me fairly often when I'm hiking. If a limited view of the sky cuts you from 4 satellites to 3 for 15 minutes, it would be nice to still be able to use GPS. However, we now have so many different satellites up there (GPS, Galileo, Beidou, and GLONASS) that with a multi-constellation receiver you should probably always have plenty of satellites visible. I think in the future consumer-grade GPS units will all start to be multi-constellation.
As Aron pointed out in a comment, it may be easier to jam a GPS signal than to destroy a GPS satellite.