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.
Aaron Carter is still in love with Hilary Duff
Millions of Getty images now available for free via embed tool