Mena said two policemen were shot and wounded on Egypt's side of the border, after witnesses had earlier reported three policemen were shot.
Thirteen more policemen were treated in hospital for injuries caused by stones, medics said.
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The Associated Press news agency reported that the tensions erupted when Egyptian guards sealed the border hermetically, not even allowing Egyptians and Gazans who had found themselves on the wrong side of the border to return home.
At some point, dozens of teens threw stones at an Egyptian command post in the area.
Egyptian security officials in nearby El-Arish said Egyptian officers fired in the air, and there were several wounded.
In Cairo, an official said there was a heavy exchange of fire.