The problem is the way NAT and DHCP work, when powering up devices, there is a certain sequence you have to follow. When everything is powered up at once, the router will try to pull a IP before what ever device that white box is is ready. So when the white box is ready, you then have to reboot the router so it repeats the process it failed at the start. PPP(user/password) can be set to auto repeat, but PPP just gets you into the network, you still have to use DHCP to get a IP. BTW, any idea what the white box is? Does it have a model number or words on it?
Allow me to simplify this. What you are describing is normal when you lose power. Is there someone else with your type of internet(dish on roof) that doesn't have to power cycle the equipment after power loss? I think the fix to the problem is either do something so you stop losing power, or put a battery backup in. It may cost more money, but considering it keeps your security up, I would think it would be worth it. Otherwise someone could cut your power to break in. What other type of solution were you looking for?