The square is the first shape invented by humans.

It symbolizes:
–the created universe
–earth and the 4 elements
–the 4 corners of the earth

It is a stable, unmoving shape.

Temples and holy buildings are square and align with the 4 points of the compass.

Squares define limits and boundaries.

From Learn Religions
Materiality – associated with material things; have solidness. Pairings with circles and squares, circles symbolize heaven and squares with earth.
Order and Stability – standing for firm foundations, both literally and metaphorically; a permanent structure that symbolizes civilization.
Spirituality – “Four” here emphasizes the four quadrants or boundaries of the earth, distinguishing it from the heavens, which are boundless.
Magic Squares —Magic squares are squares that have been broken up into smaller squares, each with a number within it, and each column and row of numbers add up to the same value. A carpenter’s square—paired with a drawing compass is a primary symbol of Freemasonry

Native Americans and The Square

 From Warpaths to Peacepipes
The Iroquois Confederacy Wampum belt contains a square symbol for each of the original five tribes in their geographic order. The first square symbol represents Seneca tribe (Keepers of the Western Door), the centre arrowhead symbol represents the Onondaga tribe, the next symbol the Oneida tribe and the end square symbol the Mohawk tribe (keepers of the eastern door). Each of the square symbols is linked by a line running between them which symbolised peace in the Confederacy.

Hiawatha Wampum Belt of the Iroquois Confederacy

The Square Symbol – Native American Meanings
The meaning of the Square symbol was used in conjunction with the enclosure sign to signify that an area had been set aside for a special event – a ceremonial dance or ritual. Native American Indians recorded their historic events through storytelling and symbolic drawings. There were so many tribes of Native American Indians it is only possible to generalise the most common meaning of the Square symbol or pattern. Native Indian symbols are still used as Tattoos and were used for a variety of reasons and depicted on numerous objects such as tepees, totem poles, musical instruments, clothes and War Paint. Indian Tribes also used their own Colors for Symbols and designs depending on the natural resources available to make Native American paint.

Mandalas

A mandala (emphasis on first syllable; Sanskrit मण्डल, maṇḍala – literally “circle”) is a spiritual and ritual symbol in the Indian religions of Hinduism and Buddhism, representing the universe. In common use, “mandala” has become a generic term for any diagram, chart or geometric pattern that represents the cosmos metaphysically or symbolically; a microcosm of the universe. The basic form of most mandalas is a square with four gates containing a circle with a center point. Each gate is in the general shape of a T

Thangka painting of Manjuvajra mandala

Mandalas in Hinduism
A yantra is similar to a mandala, usually smaller and using a more limited colour palette. It may be a two- or three-dimensional geometric composition … and may incorporate a mantra into its design. It is considered to represent the abode of the deity. Each yantra is unique and calls the deity into the presence of the practitioner through the elaborate symbolic geometric designs. According to one scholar, “Yantras function as revelatory symbols of cosmic truths and as instructional charts of the spiritual aspect of human experience”

There is also a political symbolism that you can read more about here.

Mandalas in Buddhism
There is a lot of material out there regarding mandalas in Buddhism.In Vajrayana Buddhism, mandalas have been developed also into sandpainting, where the mandala can be shown to represent in visual form the core essence of the Vajrayana teachings, that the mind is “a microcosm representing various divine powers at work in the universe. It can represent a whole universe, with Mount Meru at its center. In some, the outer circle of fire symbolizes wisdom. There are some that have Five Buddhas archetypal forms that embody various aspects of enlightenment. In tantric practice, Mandalas are commonly used by Buddhists as an aid to meditation. A “mandala offering” in Tibetan Buddhism is a symbolic offering of the entire universe. One Japanese branch of Mahayana Buddhism—Shingon Buddhism—makes frequent use of mandalas in its rituals as well, though the actual mandalas differ. The mandala in Nichiren Buddhism is called a moji-mandala (文字曼陀羅) and is a paper hanging scroll or wooden tablet whose inscription consists of Chinese characters and medieval-Sanskrit script representing elements of the Buddha’s enlightenment, protective Buddhist deities, and certain Buddhist concepts.

Mandalas in Mesoamerican Civilizations
One of several parallels between Eastern and Mesoamerican cultures, the Mayan civilization tended to present calendars in a mandala form, as in Mayan Tzolk’in. It is similar in form and function to the Kalachakra (Wheel of Time) sand paintings of Tibetan Buddhists. The tzolk’in wheel has 260 segments, surprising because the Mayans recognized that the calendar year is 365 days long. The inclusion of the specific number 260 could however relate to the 26,000 year cycle of the precession of the equinoxes.

Mayan Tzolk’in

Mandalas in Archeology
One of the most intense archaeological discoveries in recent years that could redefine the history of eastern thought and tradition of mandala is the discovery of five giant mandalas in the valley of Manipur made with Google Earth imagery. Located in the paddy field in the west of Imphal, the capital of Manipur, the Maklang geoglyph is perhaps the world’s largest mandala built entirely of mud. The site wasn’t discovered until 2013 as its whole structure could only be visible via Google Earth satellite imagery. The whole paddy field, locally known as Bihu Loukon, is now protected and announced as historical monument and site by the government of Manipur in the same year.

Mandalas in Architecture
Buddhist architecture often applied mandala as the blueprint or plan to design Buddhist structures, including temple complex and stupas. A notable example of mandala in architecture is the 9th century Borobudur in Central Java, Indonesia. It is built as a large stupa surrounded by smaller ones arranged on terraces formed as a stepped pyramid, and when viewed from above, takes the form of a giant tantric Buddhist mandala, simultaneously representing the Buddhist cosmology and the nature of mind.

Borobudur ground plan taking the form of a Mandala

LOGOS
Many financial services opt to use squares and rectangles for their logos. Wells Fargo, American Express, and Visa all use them. The shape conveys reliability and longevity, while also feeling traditional, trustworthy, and authoritative (all important qualities in that industry). Some media and technology companies also use squares and rectangles, lending an air of gravitas (NPR, BBC, and Microsoft are some examples).