Think I figured out what's going on, but I'm not sure how to fix it. Open Microsoft Band app. I open Manage Tiles from the hamburger menu and I get a pop up for Notifications saying to enable notifications for Mail, go to phone's settings, tap Accessory Apps, and turn the switch on for Microsoft Band. I tap Okay, and I can see that the listings and switches for Mail, Calls, Facebook, Facebook Messenger, Twitter, and Notification Center are grayed out. I can't change them at all in the Microsoft Band app anymore.
I go to my phone settings as the pop up suggested, but I don't see Accessory Apps. I did a search for it and it finds it. I eventually found it myself buried in the Privacy section. Opening Accessory Apps, a screen comes up and says "User consent, Turn notifications on or off for any accessory companion apps. Turning on notifications allows an app to receive your notification data, such as text messages and caller IDs." The only two things on that page are "Learn more" and "Privacy Statement" both of which also just open a couple webpages. There are no switches and no button to give consent.
Again, not sure what to do here, but I think this is what's preventing myself and others from getting updates pushed to our Bands.