Note, this isn't about the Dashboard plugin -- e-newsletter isn't showing up in the options, so I couldn't select that.
It seems that I can't do automatic updates for E-Newsletter because of the presence of a custom template. Every time I do an update, I get an error about not being able to remove the old plugin, I'm guessing because of permissions issues on the server -- that the plugin update process tries to remove my custom template will updating and then fails.
Hopefully it can be (re)designed in such a way so that original themes can be saved while doing updates. For now I'll try to remember to do it manually.