Weygers was born in Mojokerto, Dutch East Indies, where his Dutch parents, Albert Weijgers and Johanna van Leenhoff, owned and operated a sugar plantation and a hotel. His mother taught literature and several languages at a high school in Surabaya. Alex inherited his mother's linguistic talents and his father instilled a deep love of nature, design, and ecology into him as he accompanied his father on botanical explorations in Java and Indonesia.
In 1916, his prosperous parents sent him to the Netherlands to study. First he attended a secondary school where, among other things, he studied the discipline of blacksmithing—which he often referred to as the "mother craft" of all civilization. He graduated from Groningen Politechnicum in mechanical engineering and from a Dordrecht vocational university in shipbuilding. He also briefly attended the Royal Academy of Art, The Hague. Weygers returned to Java in 1923, and his fiancee, Jacoba Hutter, joined him there from the Netherlands in 1924, where they married. Jacoba could not adjust to the tropical climate, and in November of 1926 they immigrated to the United States to take residence in Seattle, where he was employed as a mechanical draughtsman. In September of 1928, Jacoba died during the stillbirth of their only child due to hemorrhaging following a Caesarean section.
Devastated by the death of his wife, Weygers decided to abandon engineering for art. In July of 1929, Alex took a summer class organized by the Art Institute of Seattle (the precursor of the Seattle Art Museum and unrelated to the identically named for-profit institution), studying sculpture under Avard Fairbanks. He created one of his most notable sculptures, Mourning, that caught the attention of world-renowned sculptor Lorado Taft in which he won a scholarship at Lorado Taft Midway Studios in Chicago. Following that he studied various aspects of art in the European centers that were renowned for the areas that interested him. Moving to California in the 1930s, he established a studio in Berkeley and began teaching. Alex was naturalized as a U.S. citizen in 1934.
In August 1942, he entered the U.S. Army and his command of Malay, Dutch, Italian, German, and English led to his assignment to the intelligence operations.
He received a patent from the U.S. Patent Office for his discopter in 1944 and his design has served as the prototype for other similar disk and hovering aircraft that have been developed up to the present day.
During his service in the army he was given a Carmel Valley property where, over several decades, he and his new wife, Marian, would build a retreat with a residence and studios, while he pursued his career teaching at Berkeley.
Marian Weygers, his second wife, had graduated from the University of California at Berkeley as an art major where she worked and studied under Chiura Obata, who taught her ink wash painting and design. She developed a printmaking process that she named "imprints from nature", using natural materials such as flowers, leaves, and grass as well as rocks and insects.
Alex and Marian Weygers relocated to the Monterey Peninsula in the 1960s and settled into their former retreat in Carmel Valley that then served as their home and studios. This was the location of his death at the age of eighty-seven. Marian remained very active in environmental and civic issues in Carmel Valley until her death in 2008.