Hi Josh, My advice would be to try to update your device via iTunes if you have a computer accessible to yourself, saying that it sounds like you are trying to do an OTA update through your iPhone. I have also had this issue, though my iPhone(s) wont even get to the software update page, as they freeze when I click on the tab. Though using iTunes to update worked. Hope this Helps! If not comment below if it doesn’t.
Sounds like your battery or a motherboard problem. I would try replacing the battery first, and if that doesn't work I would suspect other problems. All iPhones 4S and newer need a battery that is functional in order to boot.
Try putting it in dfu mode (DFU_ModeDFU MODE), connect it to your computer and restore. Your computer should detect it or try restarting all of the apple processes in task manager and reloading iTunes. Hope this helps.