You don't appear to be getting the shape reference from the Connect object you've retrieved. So how about something like this:
Set shpConn = shpStart.FromConnects(1).FromSheet
Dim vFromCellStop As Cell
Set vFromCellStop = shpConn.CellsU("EndX")
vFromCellStop.GlueToPos shpStop, 0.5, 0.5
If you're using Visio 2007 on then you might find it simpler to delete the original connector and use the AutoConnect method on the Shape object. If you're looking at a previous version or specifically want to deal with which cells glues to which then Connects and FromConnects are better.
Here's a couple of VBA examples of both methods:
Sub AutoConnectExample()
Dim shpStart As Shape
Dim shpStop As Shape
Dim vPag As Page
Set vPag = ThisDocument.Pages(1)
Set shpStart = vPag.Drop(ThisDocument.Masters("Start"), 1, 3)
Set shpStop = vPag.Drop(ThisDocument.Masters("Stop"), 3, 1)
shpStart.AutoConnect shpStop, visAutoConnectDirNone
End Sub
Public Sub ConnectWithGlueExample()
Dim shpStart As Shape
Dim shpStop As Shape
Dim vPag As Page
Dim shpConn As Shape
Set vPag = ThisDocument.Pages(1)
Set shpStart = vPag.Drop(ThisDocument.Masters("Start"), 5, 7)
Set shpStop = vPag.Drop(ThisDocument.Masters("Stop"), 7, 5)
Set shpConn = vPag.Drop(Application.ConnectorToolDataObject, 1, 1)
Dim vFromCellStart As Cell
Dim vToCellStart As Cell
Set vFromCellStart = shpConn.CellsU("BeginX")
Set vToCellStart = shpStart.CellsSRC(visSectionObject, visRowXFormOut, 0)
Dim vFromCellStop As Cell
Dim vToCellStop As Cell
Set vFromCellStop = shpConn.CellsU("EndX")
Set vToCellStop = shpStop.CellsSRC(visSectionObject, visRowXFormOut, 0)
vFromCellStart.GlueTo vToCellStart
vFromCellStop.GlueTo vToCellStop
End Sub
If you run these two you should get the following results (note - no error checking and assumes the masters exist in the document you're calling the code from):
