This seems to work:
AnimationVideo[
Graphics3D[
MeshPrimitives[
DiscretizeRegion@
RegionIntersection[Sphere[{0, 0, 0}, 2],
InfinitePlane[{0, 0, z}, {{1, 0, 0}, {0, 1, 0}}]], 1],
ViewPoint -> {2, 2, 2},
PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}], {z, -1.95, 1.95, .5}]
I really I knew an easier way to turn a Region
into Graphics3D
.