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The plane had 66 people on board. Egypt's civil aviation minister has said the possibility of a terror attack is stronger than technical failure.
Greek, Egyptian, French and UK military units have been taking part in a search operation near Greece's Karpathos island.
Greek Defence Minister Panos Kammenos said a body part, two seats and at least one suitcase had been found.
The search is now focused on finding the plane's flight recorders, the Associated Press news agency reports. On Thursday, some wreckage was reported found near the Greek island of Karpathos, but that report proved to be wrong.
POSSIBLE BREACH OF SECURITY
Three investigators from the French air accident investigation bureau, along with a technical adviser from Airbus, have joined the Egyptian inquiry.
In France, the focus is on whether a possible breach of security happened at Paris Charles de Gaulle airport.
Sherif Fathi, Egypt’s aviation minister, said he did not want to prematurely draw conclusions, but added: “The possibility of having a different action or a terror attack, is higher than the possibility of having a technical failure.”
Asked if he could rule out that terrorists were behind the incident, Egyptian Prime Minister Sherif Ismail told reporters: "We cannot exclude anything at this time or confirm anything. All the search operations must be concluded so we can know the cause."
However, there has been "absolutely no indication" so far as to why the plane came down, French Foreign Minister Jean-Marc Ayrault said on Friday morning.
“The information we have gathered confirms, alas, that this plane has crashed, and it has disappeared”, affirmed François Hollande at a press conference. “No theory is ruled out and none is certain right now.
“When we have the truth we will draw our conclusions; whether this was an accident or something else, perhaps terrorist”, the French president added.
In Washington, U.S. President Barack Obama received a briefing on the disappearance from his adviser for homeland security and counter-terrorism, the White House said.
VANISHED FROM RADAR
Flight MS804 left Charles de Gaulle airport in Paris at 23:09 local time on Wednesday (21:09 GMT) and was scheduled to arrive in the Egyptian capital soon after 03:15 local time on Thursday.
EgyptAir said the plane had been flying at 37,000ft (11,300m) when it disappeared from radar shortly after entering Egyptian airspace.
Greek aviation officials say air traffic controllers spoke to the pilot when he entered Greek airspace and everything appeared normal.
They tried to contact him again at 02:27 Cairo time, as the plane was set to enter Egyptian airspace, but "despite repeated calls, the aircraft did not respond". Two minutes later it vanished from radar.
EGYPT, A TARGET AGAIN
The aircraft was carrying 56 passengers - with one child and two infants among them - and 10 crew, EgyptAir said. They included 30 Egyptian and 15 French nationals, along with two people from Iraq and one each from Algeria, Belgium, Britain, Canada, Chad, Kuwait, Portugal, Saudi Arabia and Sudan.
With its archaeological sites and Red Sea resorts, Egypt is traditionally a popular destination for Western tourists. But the industry has been badly hit following the downing of the Russian Metrojet flight last October, killing all 224 people on board, as well as by an Islamist insurgency and a string of bomb attacks.