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A BBC correspondent in Cairo said protesters clashed with police at the hospital in the town of Naga Hamady.
The shooting happened as churchgoers left midnight Mass to welcome in the Coptic Christmas on 7 January.
The attack is thought to be in revenge for the alleged rape of a 12-year-old Muslim girl by a Christian man.
Following the reported rape in November there were five days of riots in the town, with Christian properties set on fire and damaged.
The BBC's Yolande Knell, in Cairo, said more than 1,000 Christians had gathered at the hospital to collect the bodies of six of the victims.
Stones were thrown at security forces and ambulances were smashed as they vented their anger, she added.
Three people are reported to have pulled up outside the church in Naga Hamady on Wednesday evening, killing at least six Coptic Christians and a security official and injuring 10 others, including two Muslim passers-by.
Police say the chief attacker in Wednesday's shooting has been identified but no arrests have yet been made.
The church's Bishop Kirollos said there had been threats in the days leading up to the Christmas Eve service - a reason he decided to end his Mass an hour earlier than normal.
"For days, I had expected something to happen on Christmas Eve," he told the Associated Press.
He said he left the church minutes before the attack.
"A driving car swerved near me, so I took the back door," he said. "By the time I shook hands with someone at the gate, I heard the mayhem, lots of machine-gun shots."
Witness Youssef Sidhom told the BBC that the attack shocked everyone, including police guarding the church.
Naga Hamady is 40 miles (64km) from Luxor, southern Egypt's biggest city.
Coptic Christians - who make up 10% of Egypt's 80 million population - have complained of harassment and discrimination.
Some Copts argue that previous attacks on them have gone unpunished or have resulted in light sentences.
Most Christians in Egypt are Copts - Christians descended from the ancient Egyptians.
Their church split from the Eastern Orthodox and Roman Catholic Churches in AD451 because of a theological dispute over the nature of Christ, but is now, on most issues, doctrinally similar to the Eastern Orthodox Church.