Judging by the log you don't have the battery attached. As far as I can see the modem stops responding after AT+COPS=0 command that initiates network registration, which is an indication of insufficient power supply. Please make sure the following conditions (extract from the datasheet) are met:
For powering the Boron 2G/3G version, you'll either need a USB port that is able support 2A current, or have the LiPo battery plugged in when powering over USB.
This is because the on-board u-blox modem can consumes up to 1.8A peak current when operating in 2G mode. The Boron will intelligently source power from the USB most of the time and keep the battery charged. During peak current requirements, the additional power will be sourced from the battery. This reduces the charge-discharge cycle load on the battery, thus improving its longevity.