There is a sun, there are planets, there are moons. Various ones have day/night cycles. How can you not have eclipses? Or do you mean eclipses where the sun just happens to have the same visual size of the eclipsing body like with our sun and moon?
I meant eclipses from the surface of Kerbin. I have the figures:
Kerbol radius R = 261,600 km
Mun radius r = 200 km
Kerbin - Mun mean distance d = 12,000 km
Kerbin - Kerbol mean distance D = 13,600,000 km
As R/D = 0.019 radians, and r/d = 0.016 radians, eclipses would probably be annular only. From the figures I have I believe the Mun's orbit is aligned with the plane of the ecliptic so there should be such an eclipse every orbit.
Objects do appear to cast shadows, but I'm now wondering if, say, distant objects such as the Mun do. The shadow for an annular eclipse wouldn't be as pronounced as for a total eclipse, of course.
ETA regarding the slingshot after thinking about it some more. For a 2-body encounter in the frame of reference (FoR) of the planet, the magnitude of the non-propelled spacecraft's entry and exit velocities are the same, although the direction vectors are different, of course. However, seen from the FoR of a third body, such as Kerbol, the magnitude of the entry and exit velocities would appear to differ by up to twice the speed of the planet relative to the third body - the sign of the difference and its magnitude depending on the direction of the encounter relative to the planet's orbital velocity vector. The planet also loses or gains a tiny fraction of its momentum seen from this FoR. So, multibody calculations aren't required for this aspect of the game as Sojourner
states, although they wouldn't be sufficiently precise for real probes.