The co-leaders reached the midway point of the PGA Tour's richest event with an 8-under 136 total, surviving a TPC-Sawgrass course that befuddled some of the sport's top stars.
Johnson, a former Masters champion, shot a 66 Friday. Kuchar posted a 68 and Na, perhaps known more for negatives than positives, had a 69 in a bid to make a move into golf's elite.
Na is a self-admitted slow player and has made major attempts to speed up his game. He gained attention last year for taking a 16 on a par-4 hole at the Texas Open.
The front-runners were one shot in front of tour rookie Harris English, whose 67 left him at 137.
Charlie Wi, Marin Laird, Jonathan Byrd, Brian Davis and former Players champion Adam Scott were tied for fifth at 138. Laird, who shared the first-round lead, was 10-under for the tournament heading to the par-5 16th, but he played the final three holes in 4-over.
It took a score of even-par 144 to make it to the weekend and among the notables who did so were Luke Donald (69-141), Lee Westwood (70-141), last week's winner Rickie Fowler (69-141), Phil Mickelson (71-142) and Tiger Woods (68-142).
Woods birdied four straight holes beginning at the par-3 eighth to avoid missing the cut in back-to-back weeks for the first time in his career.
World No. 1 Rory McIlroy, however, will not play on the weekend at this event for the third time in as many starts. He shot a 76 for a 148 total.
"I've got to figure out how to play this golf course," McIlroy said.
Stricker had made the cut in 49 straight tournaments, the longest current streak on the tour. McIlroy's missed cut was his first in a year.
There were 22 rounds in the 60s Friday, five fewer than the day before.