Hi atticus05,
When you say you reboot the box to turn it on, is that just to get the remote to work again. If you press the switch on the front of the box does it power on.
I assume you have tried new batteries in the remote to rule that out, is the box set to fast start or active start because some people have thought the box wasn't coming on when it has been set to eco start because it takes so long to power up.
You could also try factory reset of the 360.
It's only a partial system reset and not a complete factory reset. You are given 2 options,
Keep Recordings
Format Disk
Choosing Keep Recordings, means you won't lose your recordings and any planned recordings or series links you've set up, but you will have to sign into your apps such as iPlayer, Prime, Netflix and ITVX again, also Match frame rate and Standby power consumption will go back to their default settings if you have changed them.