Try
su minecraft -c '/bin/bash /path/to/script/script.sh &'
- The user should be the first argument to su.
- You should use quotes and not ticks for the command argument (-c)
- You may want to consider using
su -l minecraft
to have the script run in an environment which would be similar to that if the user minecraft logged in directly.
Give this a shot and let me know if it works.