Nyquist, named after Detroit Red Wings forward Gustav Nyquist, finished third at the Preakness Stakes on Saturday.
"He's out because of sickness," O'Neill told The Associated Press.
Nyquist had a temperature of 102 degrees Monday and blood work was done. He is recovering at Pimlico Race Course in Baltimore and will be shipped back home to California once he recovers.
"We just want to do right by him and get him 100 percent," O'Neill's assistant Jack Sisterson said.