Well, let's look at the wording for Technical Fighting.
Technical Fighting (AG 186)
Whenever you use a combat maneuver any damage dealt by attacks using it deal an additional 2 damage, and your maneuver DC increases by 1.
For this feature, I it depends on how you interpret "using it" to mean. I would personally interpret it that you use the stance when you activate it; though the benefits linger as long as the stance lingers, you are not using (i.e. activating, spending a cost of exertion/an action/both) the stance every turn, so the bonus damage would not apply. This means that the bonus damage would not ever be applied to stance-based maneuvers unless an attack was made as part of activating the maneuver (and even then only to that specific attack), but it would be applied to maneuvers that explicitly make an attack as part of the maneuver. I think this feature makes the most sense when interpreted this way and steps on the least toes. Otherwise, with Zealous Stance as an example, this feature provides a flat +2 damage bonus to all melee weapon attacks made while the stance is active, potentially indefinitely, which I do not think is the intention—really steps on the toes of the Dueling fighting style, for example, granting the same bonus damage to one-handed melee attacks... except without the restriction of not being able to wield a weapon in your other hand, meaning you could dual wield and get the bonus damage on your off-hand attacks. Additionally, having this as an ongoing effect would make Great Weapon Fighting pretty pointless, in my opinion, as you will have a much more consistent damage boost through Technical Fighting for the same weapons.
You used Zealous Stance as an example (AG 483), which is used (activated) as a bonus action (and does not make any attacks as part of that activation). If we count this stance as being used every single time the character benefits from the stance (getting an expertise die on melee weapon attacks), then that logic must be applied to the other stances as well: let's use Farshot Stance (AG 467) as an example. When you use this stance the range on your ranged weapon attacks increases. Should this, combined with Technical Fighting, mean that you would get a +2 damage to all ranged weapon attacks while this stance is active? After all, even if you aren't actively shooting at maximum range, you have an increased range available to your weapon every time you make an attack, which means you could say that you are using it with every attack. Just like how interpreting Technical Fighting to apply to every attack benefitting from a stance largely rendered Dueling and Great Weapon Fighting redundant, interpreting Technical Fighting this way also overshadows the Archery fighting style.
I would like to note that I don't think this feature is worded particularly clearly, but I do think a little thought and application shows it should be interpreted one way.
Let's look at maneuver specialization now.
Maneuver Specialization (AG 187)
Any attack you make as part of a mastered maneuver (including triggering attacks) deals 2 additional damage.
I think this feature is worded much less clearly than Technical Fighting—"as part of a mastered maneuver" is much more broad than "using [the maneuver]". Perhaps it was worded this way intentionally to allow every attack made while a stance was active, that was affected by the stance in some way, to deal additional damage, or perhaps this was an oversight. I don't know. That said, I would take a more lenient interpretation with this ability at my table and with my characters, largely because because Maneuver Specialization is a 3rd-level ability that already overlaps with one of the fighting stances (Technical Fighting) regardless of how strictly you interpret the above wording. (Technical Fighting on the other hand, is a 1st-level fighting style that, when interpreted too liberally, renders a bunch of other features of the same level largely redundant.)
Anyway, that's my thinking on the subject. I hope I arranged my thoughts clearly enough.
As a side note, these two fighter features stack, right? You could use Maneuver Specialization and Technical Fighting to get a +2 DC and +4 damage to certain maneuvers? Hadn't thought of that before since I don't pay a ton of attention to fighters, but this sounds like a fun combination with maneuvers that rely heavily on DCs or that allow for lots of extra attacks.
