When following the Giant Queen Bee its good to have a full set of ghillie armor, bug repellant and for some reason crouching seems to put you slightly below the drones hit range, as long as you keep moving that is. I used this tactic earlier today to tame my first queen and it worked very well
ALWAYS break the hive with ranged attacks. If you choose to melee it,drones will continuously spawn from it infinitely, and they will repair the hive, so best to shoot it down and kill all the drones before attempting a tame. Oh, and the queen will spawn more drones periodically, so bee careful.
Easiest way : go to ab then look for the hives , the hives stack so destroy all hives with a mana or whichever tame you have then the bee will be stuck in the hive , what you can do is is get an angle on the bee and easy tame while its sitting