Honestly, my evidence is that it never specifically limits which type of attack, or specifically allows it for melee basic attacks or breath weapons.
I don't have access to my books or the compendium right now, but from the character builder:
"Combat Challenge: Every time you attack an enemy, whether the attack hits or misses, you can choose to mark that target. The mark lasts until the end of your next turn. While a target is marked, it takes a -2 penalty to attack rolls if its attack doesn’t include you as a target. A creature can be subject to only one mark at a time. A new mark supersedes a mark that was already in place."
This doesn't specify which type of attack it needs to be, just "you attack an enemy."
Contrasted is the Hybrid Fighter class ability (emphasis mine) "Combat Challenge (hybrid): Every time you attack an enemy with your fighter powers or your fighter paragon path powers, whether the attack hits or misses, you can choose to mark that target. The mark lasts until the end of your next turn. While a target is marked, it takes a -2 penalty to attack rolls if its attack doesn’t include you as a target. A creature can be subject to only one mark at a time. A new mark supersedes a mark that was already in place."
Hybrid class abilities marked "Hybrid" count as the class ability, but are different in some way. The only difference between the Fighter and the Hybrid Fighter is the "with your fighter powers or fighter paragon path powers." If there was no functional difference, then it would simply state that Hybrid fighters get Combat Challenge and leave it at that. Interestingly, this means that Hybrid fighters CAN'T mark with melee basic attacks (or Dragonborn breath weapons).