Government prosecutors said yesterday they will ask a federal appeals court to overturn a ruling that evidence was obtained illegally from John W. Hinckley Jr., who is charged with shooting President Reagan and three others.

The action could delay the start of Hinckley's trial, which is now scheduled for Jan. 4.

U.S. District Judge Barrington D. Parker ruled last month that law enforcement officials violated Hinckley's constitutional rights when, on the day of his arrest, they questioned him without a lawyer present and later seized papers from his prison cell.

The prosecution is arguing that, even if the evidence were taken illegally, the government still can use that information to counter Hinckley's defense that he was insane at the time of the shooting. Prosecutors notified Parker yesterday that they intend to ask the appeals court to rule on the constitutional questions and the issue of whether the evidence can be used to rebut the insanity defense before Hinckley's trial begins.

In other developments yesterday, Parker ordered that the jury in the case be sequestered in a local hotel for the duration of Hinckley's trial to protect them from trial publicity or other outside influences.