There are tides in the atmosphere as well as ocean tides and there is also the change in speed of the earth's rotation around the sun during the new moon (which is part of every solar eclipse). A lot of animals sense impending quakes and there would be a greater susceptibility to an earthquake during an eclipse. Expect the jetstream to increase in speed.
Earthquakes do trigger one another. In fact, the modern earthquake cycle started in 1936 and accelerated with the Alaska earthquake of 1957. Each time the tectonic plates move, they cause a slight change in the location of the central axis of the earth but even though the movement is on the order of millimetres, the enormous mass of the earth makes sure that forces are generated that propagate further earthquakes.