I found a hack in modern Adobe Illustrator to get the Bezier curve handles to snap to grid points.

This worked in the 2015.0.0 Release of version 19.0.0 (64-bit):

1. While snapping to the grid with "Snap to Grid" checked under the "View" menu just make a square with the rectangle tool such that all of the corners line up on grid intersections.
2. Put one of the corners where you want a handle to snap to.
3. With "Snap to Point" also on, drag the handle's end to the corner of the square!

Ghetto!