Another year, another version. My growth as a programmer strikes me when I compare previous versions with the latest code. Less code which works better. You may compare the last three versions together on GitHub.

#!/usr/bin/env zsh

#
# Functions
#

#
# Check if Photograph is (Approximately) Square
#
# While it was originally written to detect explicitly square images, it will
# now detect squareness within a range of 20 pixels. Uses the absolute value
# of the size.
#

function is_square {
    local WIDTH=$(identify -format '%w' $1)
    local HEIGHT=$(identify -format '%h' $1)
    local SIZE=`expr $WIDTH - $HEIGHT`

    [[ ${SIZE#-} -le ${2:=20} ]]
    echo $?
}

#
# Create Directory from Date in Filename
# Images imported from Dropbox are named in format "2015-12-27 22.13.26.jpg"
#

function send_to_dated_directory {
    local FILENAME_DATE=${1:0:10}
    local DATE_REGEX='^[0-9]{4}-[0-9]{2}-[0-9]{2}$'

    if [[ $FILENAME_DATE =~ $DATE_REGEX ]]; then
        mkdir $FILENAME_DATE > /dev/null 2>&1
        mv "$1" $FILENAME_DATE
    fi
}

#
# Move File to Appropriate Location
#
# 1. To a dated folder if not an Instagram image.
# 2. To the Instagram folder otherwise.
#

function move_file {
    local EXTENSION_REGEX='^jpg$'

    if [[ ${1##*.} =~ $EXTENSION_REGEX ]] && [[ $(is_square $1) == 0 ]]; then
        mv $1 "$INSTAGRAM_FOLDER"
    else
        send_to_dated_directory $1
    fi
}

#
# Variables
#

UPLOADS_FOLDER="${HOME}/Dropbox/Camera Uploads/"
INSTAGRAM_FOLDER="${HOME}/Dropbox/Photos/Instagram/"

cd "$UPLOADS_FOLDER"

for FILE in *.*; do
    if [[ ! -f "$FILE" ]]; then
        continue
    fi

    NICENAME=$((tr '[A-Z]' '[a-z]' | tr ' ' '_') <<< "$FILE")
    mv "$FILE" $NICENAME && FILE=$NICENAME

    case ${FILE##*.} in
        gif|png) rm $FILE;;
        *) move_file $FILE;;
    esac
done