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 today.
The boy, who cannot be named for legal reasons, appeared by video-link 24 hours after he appeared at the city's youth court accused of murdering Mrs Maguire, 61, as she taught a lesson at 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.
"The consequences for individuals, I would emphasise, can be serious if there is a breach."
Richard Wright QC, defending, said there would be no application for bail.
Judge Marson said the youth will 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.