"Anyone who wanders around the world attacking IDF (Israel Defence Forces) soldiers, will not enter a school", Bennett said in a statement.
Speculation is mounting that the bird may have been launched from the Gaza Strip affixed with an incendiary device to ignite fires in Israel, a method Gazans have adopted in recent weeks with kites and balloons used as the airborne vessel to cross the border.
Meanwhile, on Monday evening, Israel announced it would again close the Kerem Shalom crossing, the only crossing for commercial goods and the main crossing for humanitarian aid between Israel and Gaza, after a rocket was sacked into southern Israel earlier in the day despite a cease fire declared by Hamas late Saturday night. "We can not stand idle when we see two million Palestinians in Gaza living in such bad conditions as they do now".
Those Israeli airstrikes were partially in response to the months of fires started by the kite firebombs, but also over continuing protests and clashes along the Gaza border.
The same day, around 200 rockets and mortars were fired at Israel from the Hamas-run enclave, and four Israelis were wounded when a rocket hit a house in the nearby Israeli city of Sderot.
The Coordinator of Government Activities in the Territories (COGAT), a department of Israel's ministry of defense that deals with Gaza, tweeted, "Arson fires have been caused by incendiary materials flown from Gaza into Israel".
Hamas on Saturday said the barrage of rockets and mortar shells into Israeli territory overnight was sacked by the "resistance" to "stop Israeli escalation".
Hamas called the initial closing last week a "crime against humanity".
Kerem Shalom is the only crossing between Gaza and Israel for the transportation of goods.
During the cabinet meeting, Israeli Defense Minister Avigdor Liberman warned Israel has "no intention of tolerating this - not rockets, not kites, not drones - nothing". The government is under pressure from local communities to show zero tolerance to the new threat from flaming kites and balloons, and Netanyahu told local leaders he has instructed the military to halt it completely. "If this is not understood through my words, it will be understood through the military's actions". Israel's Iron Dome rocket interceptor system often shoots down Gaza projectiles.
The weekend's violence came after months of near-weekly border demonstrations aimed in part to protest the Israeli-Egyptian blockade of Gaza. Over 150 Palestinians have been killed since the protests erupted along the border separating Gaza and the occupied territories in March.