They hatch from eggs, though not everywhere in the multiverse
tl;dr - In 'vanilla' 5e, they hatch from eggs, though 5e is open to different kinds of dragonborn from across the multiverse.
Vanilla 5e - they hatch
As fabio has now pointed out, 5e dragonborn hatch from eggs.
However, as the 5e books are evidently trying to be inclusive of all past D&D settings (via the "multiverse" approach), I think reference to past editions is on topic.
As first imagined, dragonborn were reborn, and their "eggs" were symbolic
The Dragonborn as such (as opposed to Dragonlance Draconians) were first introduced in the D&D 3.5 Races of the Dragon Book. Here Dragonborn are not born, but reborn from another race, and a symbolic egg is used as part of this process of rebirth:
The dragonborn children of Bahamut are a unique race in that they are not born; they are reborn. Each one enters the world as a halfling, an elf, a human, or a member of some other humanoid race with all that race’s propensities and traits. Bahamut beckons to his would-be followers, and those few who might choose to serve him.
When dawn comes, the prospective dragonborn crawls inside an egg-shaped structure that she has built ... in order to sleep for the last time in her original form. This symbolic act represents her acceptance of the transformation. When the next dawn arrives, she is transformed into a dragonborn. After awakening inside the sealed egg, she breaks out of it as a glorious, draconic child of Bahamut.
So in this original conception of dragonborn, there would be no naturally-occuring dragonborn eggs.
Dragonlance draconians (since 5e, a type of dragonborn) hatched from corrupted metallic dragon eggs
The 5e PHB, in an attempt to allow all previous D&D worlds to exist in a 'multiverse', has now labelled draconians as a type of dragonborn. As such these 'dragonborn' are created by corrupting metallic dragon eggs. This goes right back to the original Dragonlance novels and the AD&D setting.
So in this setting it would be possible to find metallic dragon eggs containing a hatchling draconian ('dragonborn').