We faced a monster that could give the condition of "nauseated" to a PC if they were hit and subsequently failed a fort save.
One of our players is a summoner who has a quadruped eidolon. Can an eidolon be nauseated? It is an extra planar outsider, and presumably does not have a stomach.
Unless the eidolon has somehow immunity to Nauseated condition, yes, it can be nauseated.
Also, Outsider template states that, while they do not require to eat or sleep, they can if they wish