Van de Zandschulp, ranked No. 59, also beat Paraguayan Adolfo Daniel Vallejo 6-2, 6-0 in the previous round. The Dutchman arrives at Winston-Salem in strong recent form, having beaten Daniil Medvedev in the second round in Montreal earlier this month.
Darderi, ranked No. 20, had climbed to a career-high No. 16 in May after saving four match points to beat Alexander Zverev in Rome, and arrived in Winston-Salem off a Santiago title this year. He could not find that level on Wednesday, managing only one game across two sets.
Bonzi, ranked No. 100, beat No. 12 seed Quentin Halys 2-6, 7-6 (5), 6-3 and Pole Kamil Majchrzak 6-4, 7-6 (7) in the previous two rounds.
Van de Zandschulp and Bonzi will now meet for a place in the semi-finals.