I read somewhere that the Draconians were born from corrupted Dragonborn eggs. I also read that they were born with some kind of additional magical ability instead of their breath weapon. If I wanted to play a Draconian how would I replace the breath with magic? The brief piece I read didn't specify the changes.

Draconians are from the Dragonlance series they are produced from corrupted good dragon eggs.

The easiest source would be to look on drivethrurpg for one of the old Dragonlance source books.  They weren't pc races so look in the monsters.  Though they weren't balanced for playing.

Draconians had magical effects that went off when they died, like turning to stone (potentially trapping a piercing or slashing weapon in their body and disarming their opponent) or exploding.

These abilities are rather obviously not useful to a player character.

Supposedly there are 5e stats for Draconians being workshopped in WotC, presumably for a Dragonlance product sometime this or next year.  Not clear whether those stats are as PC race or a monster stat block.  But yeah, like 6th said, the Draconian death effects have more utility as a DMs monster than a PC feature.  In the Dragonlance Chronicles there were spell casting Draconians, but they were clearly trained spell casters not practitioners of some innate dragon magic.  

All that said, there's nothing stopping you from basing a race on the Dragonborn, dropping the breath weapon and granting it innate Spellcasting on the line of what some other races may possess (and then design subsequent feats for "greater" magical aptitudes limited to that race or lineage).  If you ask your question in the homebrew forum, correcting from draconian to swapping out breath weapon for innate Spellcasting, someone would likely walk you through the process.