For anyone that is interested, I found the solution to my problem. Instead of setting the "AutoUpdateAfterCallBack" property in the HTML code, you would instead set it after you've updated the value for the text field, for example:
Partial Class Default
Inherits System.Web.UI.Page
Protected Sub btn1\_Click(ByVal sender As Object, ByVal e As System.EventArgs) Handles btn1.Click
txt1.Text = "Hello World"
txt1.UpdateAfterCallBack = True
Threading.Thread.Sleep(1000)
End Sub
Protected Sub btn2\_Click(ByVal sender As Object, ByVal e As System.EventArgs) Handles btn2.Click
txt2.Text = "Hello World"
txt2.UpdateAfterCallBack = True
Threading.Thread.Sleep(1000)
End Sub
End Class
That way the auto updating functionality of Anthem.Net will only update the specific field(s) that you set to update after the callback has occured.