The standard formula for the BSB upgrade is "One X (Hero) in the army may carry the battle standard for +25 points. The Battle Standard Bearer can/may carry a magic standard (no points limit). A model that carries a magic standard cannot have/may not choose any other magic items." This indicates that the battle standard is indeed equipment.
Now, HYG is a special rule and BRB Official Update Version 1.9, p. 14 specifies:
Q: If a model is transformed, for example by the Transformation of Kadon or Sivejir’s Hex Scroll, do their special rules stop working along with their magic items and equipment?(Reference)
A: No.
However, the quotes below indicate that the HYG is not tied to the BSB, but to the BS, and, therefore, it would not work, if the battle standard stops working.
BRB p. 107 : "To represent the battle standard's steadying presence, friendly models within 12" of the Battle Standard Bearer re-roll failed Leadership tests of any kind..."
BRB p. 107 (amended): “If a Battle Standard Bearer is in a unit that Refuses a Challenge and is subsequently moved to the rear of its unit, it loses the Hold Your Ground rule until the end of the turn" ("it" and "its" referring to the BS, since the BSB is elsewhere referred to as "he" and "his" - assuming that he is not suffering from gender fluidity).
BRB p. 137:" Trusted allies can use an allied battle standard's Hold Your Ground! special rule, but only if they are from the same race/realm as the allied battle standard (in other words, chosen from the same Armies book)."
BRB p. 138: "Cannot use an allied battle standard's Hold Your Ground! special rule."
BRB Official Update Version 1.9, p. 11:
Q – If a Battle Standard Bearer has to be placed in the second rank of a unit due to there being no space for them in the unit’s first rank, do the effects of the battle standard still apply? (p107)
A – Yes.
Edit (I forgot to answer the main question): the BSB does not stop being the BSB, if the battle standard's effects temporarily do not apply, because of Kadon/Svijer's Hex scroll, just like a Wizard remains a Wizard, even though he cannot cast spells etc. Hence, the BSB would still be automatically killed, if he flees from combat.
1plussave : I am not quite sure which Hydra has that effect?
rahotep75 : Slann, yes, Empire Captain (or similar) no. BRB p. 37: "Possessing a bound spell does not make a character a Wizard — he just has an item that can cast a spell."
Confirmed by BRB Official Update Version 1.9, p. 14:
Q: What happens if Sivejir’s Hex Scroll is used against a spell cast by a model without a Wizard level? (Reference)
A: Nothing, the scroll only works against models with a Wizard level.