Alan,

Convert to text is no problem.  A smooth flow may be problematic.  See the
explaination by Jay Freedman below:
Sub ScratchMacro()
'Convert TextBoxes to plain text.
Dim oShp As Shape
Dim i As Integer
For Each oShp In ActiveDocument.Shapes
  If oShp.Type = msoTextBox Then oShp.ConvertToFrame
Next oShp
For i = ActiveDocument.Frames.Count To 1 Step -1
 With ActiveDocument.Frames(i)
   .Borders.Enable = False
   With .Shading
     .Texture = wdTextureNone
     .ForegroundPatternColor = wdColorAutomatic
     .BackgroundPatternColor = wdColorAutomatic
   End With
   .Delete
 End With
Next
End Sub

Jay says:  Now the wrinkle... Every graphic object in Word's drawing layer
(e.g., a textbox) has an "anchor", a spot in the regular text to which it's
attached. (You can see the anchor symbol in the left margin of Page Layout
view if you go to Tools > Options > View and check "Object anchors", then
select a textbox or floating picture.) When you convert the textbox to a
frame and then delete the frame, the text inside gets dumped into the
regular text at the anchor position.

Good luck.

Did this document result from an OCR scan? If so, the "text boxes" may be
frames, in which case Ctrl+A, Ctrl+Q will likely remove them, but there's no
telling whether the text will be in the correct order.

If they're really text boxes, the easiest way is probably to convert them to
frames and then remove them in the same way. A macro would probably be the
best approach for doing that, but I don't know how to write it.