Boyer was born in Provo, Utah. He grew up in a nonpracticing Mormon family. He attended Provo High School, where he was active in student government and the debating team. He received a B.S. in chemistry from Brigham Young University in 1939 and obtained a Wisconsin Alumni Research Foundation Scholarship for graduate studies. Five days before leaving for Wisconsin, Paul married Lyda Whicker. They remain married and have three children: Gail B. Hayes, Alexandra Boyer and Dr. Douglas Boyer; and eight grandchildren: Imran Clark, Mashuri Clark, Rashid Clark, Djahari Clark, Faisal Clark, Lisa A. Hayes, Leah Boyer and Josh Boyer.