Quinn, a two-time Nationwide Tour winner, and Every, who will go for his first tour win Sunday, completed 54 holes at 13-under-par 200.
Chad Ginn fired a 64 to move into third place at minus-12. Esteban Toledo also posted a 64 to jump to 10-under-par 203.
Every got into the clubhouse first as he started the round tied for 11th. After an opening birdie, Every gave that stroke back with a bogey at the par-3 third.
The 24-year-old then ran off four birdies in a five-hole span from the par-4 fifth to jump to minus-9.
Every had another hot streak on the back nine. He carded three straight birdies from the 11th to move into the lead at 12 under. Every two-putted for birdie at the last to take the lead at 13 under.
'I putted great, but honestly, I didn't hit it very well,' Every admitted. 'I got some good breaks. On No. 17, I hit the trees and it kicked out in the fairway and I managed to save par.'
Quinn was three-under over his opening nine holes to make the turn at minus-10. He posted back-to-back birdies from the 12th at TPC Wakefield Plantation.
The 43-year-old tripped to a bogey on the 15th, but birdied each of the final two holes to join Quinn in the lead.
'Unforced error,' stated Quinn of his bogey on 15. 'I wasn't really thinking about getting into the last group. I was just trying to make birdies.'
Brendon Todd carded a 5-under 66 to move to 9-under-par 204. He was joined in a share of fifth place by overnight-leader Greg Owen (71), Brad Fritsch (69) and Scott Gutschewski (68).
Oskar Bergman, Reid Edstrom, Jonathan Fricke and Garrett Osborn share ninth at minus-8.