Bash if statement with multiple conditions throws an error

Use -a (for and) and -o (for or) operations.

tldp.org/LDP/Bash-Beginners-Guide/html/sect_07_01.html

Update

Actually you could still use && and || with the -eq operation. So your script would be like this:

my_error_flag=1
my_error_flag_o=1
if [ $my_error_flag -eq 1 ] ||  [ $my_error_flag_o -eq 2 ] || ([ $my_error_flag -eq 1 ] && [ $my_error_flag_o -eq 2 ]); then
      echo "$my_error_flag"
else
    echo "no flag"
fi

Although in your case you can discard the last two expressions and just stick with one or operation like this:

my_error_flag=1
my_error_flag_o=1
if [ $my_error_flag -eq 1 ] ||  [ $my_error_flag_o -eq 2 ]; then
      echo "$my_error_flag"
else
    echo "no flag"
fi