McHale, the fifth seed, defeated Jarmila Gajdosova 7-6 (7-5), 7-5. Scheepers, the eighth seed, topped Coco Vandeweghe 6-2, 7-6 (7-4).
American Vania King eliminated Kai-Chen Chang 6-1, 6-4 and Yung-Jan Chan of Taiwan advanced when seventh-seeded Yanina Wickmayer retired trailing 7-6 (8-6) after one set.
New Zealand's Marina Erakovic, Hungarian Melinda Czink, Britain's Heather Watson and American Melanie Oudin also posted wins Tuesday.
2014: The Year in Music [PHOTOS]