Reynolds, 55, of Grand Blanc, Mich., had advanced to the third round with a 2-up victory over Mike Riley of San Diego in Tuesday's morning round.
Clagett, 55, of Austin, Texas, extended his hopes of being the first medalist to win the title since 1987 with a 6-and-5 victory over Robert Blomberg of Alameda, Calif., in the second round.
Also advancing to Wednesday's quarterfinals were: Vinny Giles, of Richmond, Va., the 1972 U.S. Amateur champion, who beat Mike Rice, of Houston, 2 up; Bill Baloh, of Herminie, Pa., a 1-up winner over 2003 USA Walker Cup captain Bob Lewis Jr., of Pepper Pike, Ohio; and Alex Tarumianz, of Lookout Mountain, Tenn., who beat Bruce Walters, of Pikeville, Ky., 1 up.
Two 61-year-olds advanced to the quarterfinals ' Bill Heldmar, of Tulsa, Okla., and Curt Madson, of Mount Dora, Fla. ' as well as Mark Bemowski, of Mukwonago, Wis., and Gary Koprivica, of Missoula, Mont., who advanced to match play via a playoff.
The semifinals and 18-hole final are Thursday.
Reynolds, who made the field as an alternate four days before the championship began, lost the first hole to Clagett but came back to win the next four, two with birdies and two with pars.
'Billy starts out with a birdie on the first hole, so I thought, 'Well, maybe we're going to need to make some birdies here today,''' Reynolds said. 'The first couple of matches, you can win some holes by just making pars but by the time you get around to the third or fourth match, you've got to make some birdies too.''
After Reynolds birdied the par-5 12th to go 4 up, Clagett won the par-4 14th and par-5 15th to cut the deficit in half. They halved the par-3 16th with pars and both birdied the par-4 17th, giving Reynolds the victory.
Clagett had not three-putted in the championship before doing so twice against Reynolds.