The trio added: "When RSY-Netzer discovered that most of those killed in Gaza were claimed as Hamas operatives, they acknowledged that had they known, they would not have shared this on Facebook".
Israeli Intelligence Minister Israel Katz said his country was "at the closest point to the threshold of war" in Gaza since the seven-week Operation Protective Edge in July and August 2014.
Israeli cabinet minister Arieh Deri told Israel's Army Radio that he expected calm to be restored.
Adva Klein of Kibbutz Kfar Aza said she only got about two hours of sleep because of the frequent incoming fire and the warning sirens.
Three members of Islamic Jihad were killed in an Israeli strike on Sunday, and the group vowed revenge.
A picture taken on May 29, 2018 shows fishing boats carrying a group of Palestinian activists who are protesting and perhaps trying to breach Israel's naval blockade on Gaza, setting to sail from Gaza City harbour. Israel said most were intercepted by its air defense systems.
"Such attacks are unacceptable and undermine the serious efforts by the worldwide community to improve the situation in Gaza", he said.
The United States called for an emergency meeting of the United Nations Security Council to discuss the attack on Israel.
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The United Nations Security Council is expected to meet Wednesday to discuss the violence, following a U.S. request for an urgent meeting.
Meanwhile, Kuwait on Tuesday circulated a watered-down draft UNSC resolution on setting up an worldwide protection mission for the Palestinians in a bid to win European support in a vote expected this week, diplomats said.
Shortly after he spoke, Israel's military began air strikes in Gaza. Explosions shook the Palestinian enclave and smoke rose from areas hit. "I don't think serious people in Israel want to reoccupy Gaza", said Yaar. It said some of the mortars fired were supplied by Iran.
"Qassam and Jerusalem Brigades announce joint responsibility for bombarding [Israel's] military installations and settlements near Gaza with dozens of rocket shells throughout the day", they said in a joint statement.
"We must prepare to occupy Gaza as a part of efforts to eliminate Hamas rule", Shaked said, according to a translation by Palestinian-Israeli news website Arab48.
Later Jihad spokesman Dawoud Shihab said an Egyptian-brokered agreement had been reached to return calm. Israel and the US have described Hamas and Islamic Jihad as a terrorist organizations. In the initial barrage on Tuesday morning, a rocket hit a kindergarten, although no children were inside at the time.
"The Israeli occupation bears full responsibility for any upcoming escalation", it said. "Fifteen boats attempted to cross more than nine nautical miles (16.7km) but met open fire by Israeli forces".
Tuesday's incidents came after weeks of deadly demonstrations and clashes along the Gaza-Israel border, beginning on March 30.