Are bosses guaranteed to drop two yellow items the first time I kill them on normal?

Solution 1:

It's true... sort of.

The rare drop rate is substantially higher for first time kills of major bosses on normal mode, but it isn't necessarily two drops guaranteed - the specific bonus and which bosses it applies to hasn't been laid out explicitly. For what confirmation does exist, see this post by Bashlok:

The bonus to drops for first-time boss kills only exists in Normal difficulty.

Whether the 'bonus to drops' is 'guaranteed class-specific yellow', or simply 'twenty percent increase to the chance to drop' is unclear, and without a very large data set or some (very unlikely) additional candor on the matter from Blizzard, it's difficult to say. Anecdotally, people seem to be citing the '2 rares, one for your class' fairly widely, and I'm yet to personally have any experiences that contradict that - at least on major bosses. If I had to guess, I'd wager the Skeleton King doesn't qualify as one however.

Solution 2:

I am not sure Skeleton King is considered a "boss", in fact it does not appear in Highest Progress. The Butcher is the first boss and it should indeed drop at least 1 rare (yellow) item.