Based on: http://nicknotfound.com/2013/03/21/switching-from-rvm-to-rbenv-on-a-production-server/

puma.sh

#!/bin/sh

### BEGIN INIT INFO
# Provides:          puma
# Required-Start:    $all
# Required-Stop:     $all
# Default-Start:     2 3 4 5
# Default-Stop:      0 1 6
# Short-Description: starts the puma app server
# Description:       starts puma using start-stop-daemon
### END INIT INFO

set -e

USAGE="Usage: $0 <start|stop|restart|upgrade|rotate|force-stop>"

# app settings
USER="admin"
APP_NAME="myapp"
APP_ROOT="/home/$USER/apps/$APP_NAME/current"
ENV="production"

# environment settings
PATH="/home/$USER/.rbenv/shims:/home/$USER/.rbenv/bin:$PATH"
CMD="cd $APP_ROOT; bundle exec unicorn -c config/puma.rb -E $ENV -D"
PID="$APP_ROOT/tmp/pids/puma.pid"
OLD_PID="$PID.oldbin"

# make sure the app exists
cd $APP_ROOT || exit 1

sig () {
  test -s "$PID" && kill -$1 `cat $PID`
}

oldsig () {
  test -s $OLD_PID && kill -$1 `cat $OLD_PID`
}

case $1 in
  start)
    sig 0 && echo >&2 "Already running" && exit 0
    echo "Starting $APP_NAME"
    su - $USER -c "$CMD"
    ;;
  stop)
    echo "Stopping $APP_NAME"
    sig QUIT && exit 0
    echo >&2 "Not running"
    ;;
  force-stop)
    echo "Force stopping $APP_NAME"
    sig TERM && exit 0
    echo >&2 "Not running"
    ;;
  restart|reload|upgrade)
    sig USR2 && echo "reloaded $APP_NAME" && exit 0
    echo >&2 "Couldn't reload, starting '$CMD' instead"
    $CMD
    ;;
  rotate)
    sig USR1 && echo rotated logs OK && exit 0
    echo >&2 "Couldn't rotate logs" && exit 1
    ;;
  *)
    echo >&2 $USAGE
    exit 1
    ;;
esac

I'm really suspicious of the line in this script

   ENV="production"

I'm pretty sure it should be

   RAILS_ENV="production"

I used a similar script, and everything seemed fine, but rails had no idea it was production because the environment wasn't set correctly.