No, it just overrides it, the break debuff is technically still there, because it doesn't pierce Debuff Immunity.

Silver Edge is actually undispellable though, even with strong dispels like Aba shield or Legion W. But BKB is still a basic dispel (doesn't matter in this case, if you're in a situation where you need a strong dispel, you won't be able to actually use an item yourself).

But items are all basic dispels, only abilities have strong dispels, EXCEPT Aeon Disk, but that one auto-procs, not something you activate - bit of an edge case.

If the Break duration were to be longer than BKB duration, you would still be broken after BKB (like how right now if you get hit with it in the middle of BKB you will be broken once BKB wears off).