I click the button, but nothing happens

Point 1: I assume you're really using one of the official releases, not a -dev release.

That said, sorry, there isn't much here for me to work on here. There must be something funny with your configuration, but I can't think what. You're right, you shouldn't see a button to sign up if you can't actually sign up, but I haven't seen that before.

Could you check the site's error logs and see if an error is being reported somewhere?

If that doesn't yield anything, I'd be willing to take a quick look at your site if you send me a pointer. I'd recommend that in a private message, don't put that here unless you want the world to see it.

It appears that this is an issue with the Bootstrap theme that you're using. I don't know much about themes, or why the "collapsable fieldset" that DateReminder uses doesn't work in that theme, but I see the same behavior with the Subscriptions module.

I suggest that you use the "As a tab" display option with this theme.

A workaround was added in release 7.12: There is now a configuration option to automatically expand the fieldset in the node. By default, the fieldset is displays initially as collapsed. But if you unselect the option, the form will always appear. That makes the form appear when the theme doesn't support collapsable field sets. (It turns out there was already code to support the option, just no way to set it.)