Answer: 137

Small Hint:

First find the distance between the two centers

Big Hint:

The maximum occurs along the line of centers, on the two outward sides

Solution:

The squared distance between the centers is $14^2+18^2+(-21)^2=961,$ so they are $\sqrt{961}=31$ apart. By the triangle inequality, the greatest point-to-point distance is obtained on the line of centers and equals $19+31+87=137.$