Judge Geoffrey Marson QC said serious consequences could follow for individuals who named the teenager online or published other information covered by reporting restrictions.
Judge Marson made the comments as he remanded the youth in custody at Leeds Crown Court yesterday.
The boy, who cannot be named for legal reasons, appeared by video-link 24 hours after he went before the city’s youth court accused of murdering Mrs Maguire as she taught a lesson at Leeds’s Corpus Christi Catholic College on Monday morning.
The judge said journalists generally understood the restrictions that applied in the case, but added: “What may not be understood by many is that these constraints and prohibitions apply also generally, including users of online social media sites and bloggers.
“They are just as much bound by these constraints as are the Press. The consequences for individuals, I would emphasise, can be serious if there is a breach.”
The boy appeared in court on two large video screens from the centre where he is being held.
Neither his parents nor any family of Mrs Maguire were in court.
Dressed in a purple and yellow halved sweatshirt and matching trousers, the teenager stared at the floor for part of the 15-minute hearing but also looked straight at the camera, occasionally pushing his fringe off his face.
Richard Wright QC, defending, said there would be no application for bail.
Judge Marson said the youth would next appear at Leeds Crown Court for a plea and case management hearing on July 11 and fixed a provisional trial date for November 3.
Mr Wright, prosecutor Paul Greaney QC and the judge all sat in court without their normal wigs and gowns and dispensed with some of the normal protocols of the Crown Court due to the defendant’s age.
Meanwhile, police have confirmed that a second teenager was arrested in connection with the fatal stabbing.
West Yorkshire Police said the boy was arrested on Thursday and later released without charge.
The force would not confirm what offence he was arrested on suspicion of, but said it was not murder.
Mrs Maguire’s death is the first time that a teacher has been stabbed to death in a British classroom.
She originally went to Corpus Christi as a student teacher and last year the school held a celebration of her 40 years of service.
This year, she moved to working four days a week ahead of her planned retirement in five months.
Mother-of-two Mrs Maguire lived in the Moortown area of Leeds with her husband Donald, who is a retired maths teacher and landscape gardener.
A sea of floral tributes has built up outside the Corpus Christi school, which is in Halton Moor. A book of condolence has also been opened at nearby Corpus Christi Catholic Church.
Comment: Page 14.