Your AstroBox will only start a hotspot when it doesn't find a network that it knows. If you have successfully configured your network connection during setup, it will remember it and connect to it while in range.
After a reboot, it will find your network and determine that it's no longer needed to start a hotspot. It might seem that your AstroBox has disappeared or having problems booting up. However what's happening is that your AstroBox is now registered to you local network.
Simply use a device that's connected to the same network you configured and navigate to http://astrobox.local on it. If you changed the name of your AstroBox use http://<name>.local.
Note that in Windows machines, .local addresses won't work right away. You need to install Bonjour support as explained here.
Let's say you have this scenario:
You named your AstroBox "DrewsPrinter."
Your home wifi network is called "Home Sweet Home Wifi"
Now you turn on your Astrobox. If your AstroBox has ever connected to the Home Sweet Home Wifi network, it will remember that and will connect to it automagically! You connect your phone, tablet, or pc to Home Sweet Home Wifi and you can now connect to the AstroBox. Your AstroBox will also show up in your AstroPrint.com connected devices panel. The AstroBox will not create a hotspot, because a hotspot is not necessary.
If your AstroBox cannot find any networks that it knows, then it will create a hotspot named DrewsPrinter. Then connect your phone, tablet, or pc to DrewsPrinter wifi network and go to DrewsPrinter.local, or http://10.10.0.1. You are now connected to your AstroBox.