Depends on the house rules. Seems like it could be a bad beat to me, but my opinion doesn't mean squat.
I do need to know if both hole cards are played - I.e. AA vs KK preflop. If this is AQ vs KK on a board like < AAA > K K then the wining hand is four aces, king kicker making the queen in hand not playing.
How often does any of this happen without wild cards anyway?? BTW this is a home game forum so BBJs are what ever the host and players say it is I would think.
Quad Kings being beaten by a Royal seems a tad worse than Quad Queens. Not that this kind of hand is very likely to happen without wild cards anyway. If I was host , and had to rule on this fantasy event , I would certainly award the BBJ to the player with Kings ! Home game structures with BBJs will vary a whole lot no doubt!
Most places will want pp for bad beats. Is the question about the KK rivering quads b/c it wasn't ahead and didn't get 'caught'? If so, that doesn't matter.
Regarding 3 bb qualifying hands at once:
Best hand (RF) wins the pot
2nd best hand (KK) is the bad beat loser
3rd best hand (QQ) isn't any more special than the rest of the table