Organisers say the break-downs happened "under suspicious circumstances".
The plan is to bring some 700 protesters and 10,000 tonnes of supplies into the blockaded territory. Israel has said it would not allow the ships to dock and has prepared warships to intercept the fleet. It says the flotilla would be breaking international law.
The organisers of the “Freedom Flotilla” describe themselves as a coalition of non-governmental organisations.
It was due to set sail from Cyprus at 09.00 universal time, but organisers were negotiating with Cypriot authorities.
They have accused Cyprus of conceding to Israeli pressure and reneging on an agreement to let them sail from there.
European Union foreign policy leader Catherine Ashton urged "restraint and responsibility" from Israel and the so-called Freedom Flotilla, as ships set sail from Cyprus en route to Gaza City.
On Friday, the EU called for an immediate end to the Israeli blockade of Gaza, which Israel has blockaded since the Islamist movement, Hamas, took control in 2007.
"The EU remains gravely concerned by the humanitarian situation in Gaza. The continued policy of closure is unacceptable and politically counterproductive," said Ashton.
Meanwhile, the Israeli military says its aircraft fired missiles at a metal workshop and a tunnel in Gaza. Hamas officials said Saturday that nine missiles were fired, but that no one was hurt.