You'll want to create your own property on the TextBox, say "Value" or "MyText" (something other than Text since it's already taken). Make that property a DateTime property and that property will then be used for your binding (so, bind the business object to that property, not the Text property.
In the Get of that property, if the Text is empty, then return 1/1/1800, if the text has a date, then use DateTime.Parse() to get the DateTime value from the Text and return it.
Then, in the Set of the property, if the value is 1/1/1800, set the text to String.Empty. Otherwise, set the Text to value.ToString("MM/dd/yyyy") (where the MM/dd/yyyy is the format you want the dates to appear in).
Luckally, WebForms binding is much simpler than WinForms binding, so you won't have to worry about creating all of those changed and changing events to get the data to bind properly.