There were no early reports of deaths or injuries at the five tented camps which have sprung up on the Palestinian side of the 25-mile (40km) border fence beside which Gazans launched what they call "The Great March of Return" on March 30.
The AFP news agency reported that another four Palestinians had been killed during the same time period, outside of the scope of the protests.
Palestinians headed into the seventh week of demonstrations as part of the "Great March of Return" on Friday, as protest organisers chose to highlight the plight of workers in the Gaza Strip. Army gunfire has killed at least 43 Palestinians on the frontier over the last month.
As Israel celebrates its 70th birthday, Palestinians mourn what they call the "Nakba" (Catastrophe) of their people's mass-dispossession during the conflict that broke out in 1948.
The protesters were calling for the right of Palestinian refugees and their descendants, who according to the United Nations make up around two-thirds of Gaza's population, to return to their homeland inside what is now Israel.
Aser was shot east of Gaza City on April 27, Health Ministry spokesman Ashraf Al Qudra said.
Anas Abu Asr was wounded near Gaza City last Friday, the fifth of weekly protests led by the ruling militant Hamas group.
An added focus this year is President Donald Trump's decision to begin moving the U.S. Embassy to Jerusalem on May 14, the 70th anniversary of Israel's founding.
"The Palestinian protests on Gaza's border fall into the "state of war" category; thus human rights laws do not apply to the [Israeli army's] rules of engagement in this regard", state attorneys asserted in response to petitions filed with Israel's High Court by a number of human rights groups.
The move has deeply angered the Palestinians, who see the Israeli-annexed eastern sector of Jerusalem as the capital of their future state.