We have been adventuring, and in our latest quest, we came upon a witch that had a couple of Draconic eggs in a cooking pot. They seemed intact, and a blue one even gave off an electrical sensation.
The guess was that they were Dragon eggs. However, as there are many Draconic-type races in 5e, there's a possibility it could be anything from Kobold to Dragon.
My question though is: does this selection include Dragonborn? I.e. Do Dragonborn come from eggs?
In 5e dragonborn come from eggs, PHB pg 34:
Age. Young dragonborn grow quickly. They walk hours after hatching
BTW you should tell apart dragon eggs from dragonborn eggs from the dimension. The dragon eggs are 4 feet tall.
In 4e the same like pointed in dragon magazine 365 pg 28:
Like true dragons, however, dragonborn hatch from eggs, usually laid singly or, more rarely, in a pair.
and then in Player's Handbook Races - Dragonborn pg 19:
From the moment you emerged from your egg...
Sort of. But probably not in the sense that you mean. From the PHB entry on dragonborn:
Shaped by draconic gods or the dragons themselves, dragonborn originally hatched from dragon eggs as a unique race, combining the best features of dragons and humanoids.
So the dragonborn as a race definitely originated from eggs. But the word "originally" there implies that they don't, in the present, hatch from eggs. It's not nearly as strong a statement as I'd prefer. It does also say this:
The first dragonborn had scales of vibrant hues matching the colors of their dragon kin, but generations of interbreeding have created a more uniform appearance.
So while dragonborn originally came from dragons, new dragonborn come from other dragonborn. That doesn't prove that they don't lay eggs, of course, but it does mean that they reproduce in a different way to their original ancestors.
tl;dr - In 'vanilla' 5e, they hatch from eggs, though 5e is open to different kinds of dragonborn from across the multiverse.
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.
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.
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').
PHB 32:
Shaped by draconic gods or the dragons themselves, dragonborn originally hatched from dragon eggs as a unique race, combining the best attributes of dragons and humanoids.
PHB 34: (Dragonlance specific, but still instructive)
In the Dragonlance setting, the followers of the evil goddess Takhisis learned a dark ritual that let them corrupt the eggs of metallic dragons, producing evil dragonborn called draconians.
Based on these conclusions, the original dragonborn were hatched from eggs. I couldn't find anything in 5e about whether dragonborn continue to reproduce by laying eggs, but I think it's a safe guess that they do. Alternatively, the eggs you came across could have been normal dragon eggs exposed to a ritual or influence that would change them into dragonborn (or maybe anything else dragon-related?), especially given that you found those eggs in a witch's cooking pot...
As the other answers point out, it is unclear if dragonborns give live birth or lay eggs. So it's up to the interpretation of the DM how it works in your campaign world.
When at least one of your characters has a background which implies that they might be familiar with dragonborn reproduction, you should just ask the DM how it works in your campaign world and if any of the eggs you encountered could plausibly be dragonborn eggs. When it is not obvious that your character would or would not know that, your DM might demand a successful intelligence check before giving your character that information.
