Andy,

The following deletes the section break immediately above the insertion
point.  You could easily adapt it to delete the one after.

Sub DeleteSectionBreak()
Dim i As Integer
Dim j As Integer
i = 1
j = Selection.Sections(1).Index
If i < ActiveDocument.Sections.Count Then
Selection.GoTo What:=wdGoToSection, Which:=wdGoToAbsolute, Count:=j
Selection.MoveLeft
Selection.Delete
End If
End Sub