Compute minimal string segment sizes
Pyth, 20 18 bytes
hljb-{.zsmteM./d.z

Demonstration.
{ can be removed if duplicates are not allowed.
Explanation:
hljb-{.zsmteM./d.z
                .z     The input, as a list of strings.
         m             Map each strings to
             ./d       all possible partitions of the string into separate strings.
           eM          take the last element of each, giving all suffixes.
          t            Remove the first suffix, giving all suffixes other than
                       the string itself.
        s              Sum, combining the list of lists into a single list.
    -{.z               From the set of input strings, remove all suffixes.
                       This is the list of strings in the minimal segment.
  jb                   Join the strings together on newlines.
 l                     Take the length of the resulting string.
h                      Add one and print.