Fifth-seeded Eubanks needed courtside treatment for a glute muscle injury and lost 7-6 (7-5) 6-2 to Dutchman Botic van der Zandschulp on Monday.
It is not yet clear whether the injury will affect Eubank's Australian Open campaign, which begins in Melbourne on Sunday.
The big-serving American indicated he was in trouble late in the first set and called for the tour trainer.
He received treatment between sets and elected to play out the match. However, he looked limited at times and seemed unable to run at full stretch.
Van der Zandschulp converted his third break point in the second game of the second set to lead 2-1, then broke again at 5-2 to seal the match.
"I'm pretty happy with my performance. I think I played a decent level overall," van der Zandschulp said.
"Chris is serving amazing and you have to keep up every time.
"If you don't pay attention for just a couple of seconds you can be broken, and it's tough to break him."
Earlier on Monday, Germany's Daniel Altmeier defeated American Marcos Giron 7-6 (7-5) 6-2, and Frenchman Luca van Assche beat American JJ Wolf 6-4 6-2.
French veteran Gael Monfils suffered a three-set defeat in the first round in Auckland. (Dean Lewins/AAP PHOTOS)
French veteran Gael Monfils was also beaten in the first round, losing 6-2 6-7 (6-8) 7-6 (7-4) to Fabian Marozsan of Hungary.
Monfils, 37, at times looked near exhaustion in the latter part of the encounter, slumping over a chair between points as the match dragged on against an opponent 13 years younger. Yet he kept coming back to win clutch points.
He served for the match at 5-4 in the second set, but Marozsan saved and rallied to win the match in the final tiebreaker.
Monfils spent all of last week courtside in Auckland, following his wife Elina Svitolina as she reached the final of the women's Auckland Classic before losing in three sets to defending champion Coco Gauff, of the US.
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