I have attached a drawing containing the connectors.
Dynamic-Connectorvsdx.vsdx

I would start with the Number Shapes Add-On in Visio to see whether it will do what you want:
Select View>Macros>Add-Ons>Visio Extras>Number Shapes.

A couple of caveats:
1) The default settings don't number shapes as you drop them onto the page.
2) The default settings exclude connectors.
3) The default settings may add numbers to shapes you don't want numbered.

You can resolve #1 by checking the highlighted box shown at the bottom of this image. The highlighted radio button at the top of the dialog lets you click on existing shapes to number them; the second setting is part of the answer to restricting which shapes are numbered (the rest of the answer is below).You can resolve #2 and #3 by using the settings show in the image below. The Selected layers setting takes advantage of the fact that all dynamic connectors in Visio are automatically added to a layer called "Connector".

One other thing to note about the Number Shapes add-on: you can include a text prefix. After adding a number to a shape, take a look at the shape data fields that are added to each shape. Type any value into the field named Shape Number Text and that text will precede the number. Granted, you will need to do this manually for each shape, but you could write code to automate this action if you want to.

I don't know whether this built-in add-on will be sufficient for your needs, but let me know what you think after you experiment a bit.

Aaron -- I'm glad to see that you did what I almost suggested yesterday, which is posting this question on John Goldsmith's blog. He's a great guy and I happen to know from private communications that he's looking into the issue...

Hello Scott,

Thank you very much for explaining the auto-numbering add on. I appreciate your assistance.

John Goldsmith has replied to my question on his blog. I will use the information that you have provided and Johns comments to create an auto-numbering dynamic connector.

Again, thank you for your assistance, I appreciate it.

I'm glad we were able to help.

One correction to my initial response above (you may have already figured this out, but for completeness, here's the correction):
In the first dialog box above I suggested you choose "Select shapes" rather than "All shapes". That's incorrect. You'll want to choose "All shapes". The "Selected shapes" option only appears if there are one or more shapes selected at the time you invoke the Number shapes add-in and will be grayed out otherwise. The option that really does what you want -- limiting numbering to Connectors -- takes place in the second dialog box by using layers.

Feel free to leave this question open a bit longer if you think you'll need additional assistance or if you'd like to post your eventual solution (which is always a bonus for someone who comes along after you looking for something like this). Otherwise, feel free to close the question.