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I've been off site for a while and catching up. Instead of reboot commands have you tried shutdown -r 03:00 as an example instead? In this example it will shutdown and reboot the system at 3 in the morning and you can change the time to whatever you want. If you did a -r 10 as example it'll reboot in 10 minutes. If you want it later in the evening shutdown -r 18:00 as example. You may be able to run this in the same script as the miner (haven't tested) or run as a separate script and have it in Cron.
*** Edit ***
Ok tested and this will work:
#!/bin/sh
echo "System reboot at 3am"
shutdown -r 03:00
sleep 5
./bfgminer --scrypt -o stratum+tcp://miningpool.gravio.net:3333 -u siteusername.worker -p x -S ALL --set MLD:clock=700
In the above script it will reboot at 3am just as a basic generic example. And since you are doing OS updates you may want to run that in a separate script. This example is a different miner but logic is the same. Possibly putting the shutdown command in it instead of the mining script.