Place n/4 balls separated along a horizontal line L1, and another n/4 along a parallel line L2 below, with each of the lower balls directly below an upper ball with their centers 1 unit apart. Thus each pair of balls overlap, their surfaces intersecting in a circle. Arrange a second set of n/4 pairs of intersecting balls along lines L3 and L4, far from L1/L2 and with all four lines parallel, and such that all circles of sphere intersections are coplanar. Now it is easy to see that a line tangent to two circles of intersection, one from the L1/L2 group, one from the L3/L4 group, is tangent to four balls. And there are Ω(n2) such lines. (The same bound can be achieved with disjoint balls with a similar arrangement, but the analysis is slight more complex.)
The problem is also interesting if all balls are disjoint; it is not clear if disjointness affects the answer asymptotically.