I believe Sears have it right. Both St. Thomas the Apostle and St. Joseph have an association to the square. In Joseph’s case it’s a
carpenter’s square since that was his profession and he has long been the patron Saint for that profession. However, I think only St. Joseph also has association with the lily.
This from the website of St. Joseph’s University in Philadlphia:
In 1955, an additional feast day was declared by Pope Pius IX and is celebrated on May 1 as the Feast of St. Joseph the Worker. This date is also International Workers’ Day and believed to reflect Joseph as the patron saint of workers. The statue in front of Barbelin Hall of St. Joseph the Worker was donated by students of The Evening School who studied in the evenings after work.
Symbols associated with Joseph include the
carpenter's square and lily. The carpenter’s square not only symbolizes Joseph’s trade but also the concept of truth. The lily, which can also be seen in the Saint Joseph’s University logo over the letter J, represents purity and Joseph’s celibate marriage to the Virgin Mary.
Joseph is the patron saint of the Universal Church, families, fathers, expectant mothers, travelers, immigrants, house sellers and buyers, craftsmen, engineers and working people, among others. In the grotto behind the Chapel of Saint Joseph stands a statue of St. Joseph the Dreamer. Serving as a space for personal retreat, prayer and reflection, the grotto features a light that is always on.
https://www.sju.edu/news/10-facts-about-st-joseph-honor-his-feast-day
This is the statue referred to, with Joseph holding a huge square: