Using if elif fi in shell scripts

sh is interpreting the && as a shell operator. Change it to -a, that’s [’s conjunction operator:

[ "$arg1" = "$arg2" -a "$arg1" != "$arg3" ]

Also, you should always quote the variables, because [ gets confused when you leave off arguments.

Josh Lee's answer works, but you can use the "&&" operator for better readability like this:

echo "You have provided the following arguments $arg1 $arg2 $arg3"
if [ "$arg1" = "$arg2" ] && [ "$arg1" != "$arg3" ]
then 
    echo "Two of the provided args are equal."
    exit 3
elif [ $arg1 = $arg2 ] && [ $arg1 = $arg3 ]
then
    echo "All of the specified args are equal"
    exit 0
else
    echo "All of the specified args are different"
    exit 4 
fi