Fox was forced to take a drop after sending his tee shot into the native area on the par-four sixth, then dropped another shot to sit two-over through six.
Two birdies and two bogeys through his remaining 12 holes saw him finish the day two-over, joining Hillier in a tie for 85th after day one.
Hillier was left to rue a tough back nine, which included four bogeys, two of which came in his final two holes.
It was the opposite for Kobori, who had a mid-morning tee time. He, too, was over par at the turn.
While he had two bogeys on the back nine, he also had four birdies to end the first round where he started, with even par putting him in a tie for 39th.
“I struck the ball really, really well,” he said. “But a few mishaps on the greens. They’re tricky. But all in all, even par, I’m pretty happy about out there.”
American Jackson Suber led the field after the first round at five-under, holding a one-shot lead over Sungjae Im of South Korea and England’s Dan Brown.
Nine players were tied at three-under, while defending champion Scottie Scheffler was among the 11 players tied for 13th at two-under.
Tee times for Kiwis at The Open Championship
Daniel Hillier (+2): Round 2 – 10.52pm Friday NZT, 11.52am Friday local time.
Ryan Fox (+2): Round 2 – 11.03pm Friday NZT, 12.03pm Friday local time.
Kazuma Kobori (E): Round 2 – 2.48am Saturday NZT, 3.48pm Friday local time.