The blade guide is not a creature – it is a projection
The blade guide isn’t a creature in the first place. It’s never actually called a creature, nor given any type. It is called a “spirit,” and it is that based on the definition of spirit given in Complete Divine, but that definition does not require that the spirit be a creature.
Furthermore, “creature” is well-defined, and the blade guide doesn’t qualify:
Anything with no Wisdom score is an object, not a creature. Anything without a Wisdom score also has no Charisma score.
Anything with no Charisma score is an object, not a creature. Anything without a Charisma score also has no Wisdom score.
[The blade guide] lacks [...] ability score modifiers.
Since the blade guide does not have Wisdom or Charisma, it is not a creature. So what is it? Tome of Battle has that answer, too:
The guide is the physical manifestation of a spirit that resides on the Outer Planes.
Thus, we see that the guide does fit the definition of “spirit” given in Complete Divine – specifically the fourth bullet point, “Creatures in astral form or with astral bodies.” The blade guide isn’t, itself, a creature, it is the projection or manifestation of one. It’s just a supernatural effect.
For anything that can possibly happen to a blade guide, there is one of two results
Either blade guide ignores the effect, as it does not apply to the blade guide, or it is destroyed and reappears 1d6 rounds later in the eternal blade’s space. There is literally no other possible result.
Undead are creatures – the blade guide isn’t undead
Since the blade guide isn’t a creature, it can’t be undead. Since it’s not undead, the hunter of the dead has no special power over it.
Outsiders are creautres – the blade guide isn’t an outsider
Again, not a creature, so not an outsider. Therefore banishment does nothing to it. Nor does the anti-summoning effects of protection from evil or magic circle against chaos or whatever.
The blade guide does have a space, and could be trapped
The blade guide has a size, a movement speed, and so on. That being the case, it can be separated from the eternal blade, and trapped.
This just doesn’t really accomplish much:
If your blade guide is destroyed, or it somehow moves out of your line of effect, it reappears in your space in 1d6 rounds.
So as soon as the blade guide enters the trap, which presumably blocks line of effect to the eternal blade, it disappears. It doesn’t leave the trap, the effect just ends; it’s destroyed. But it returns 1d6 rounds later, in the eternal blade’s space.
The blade guide is a Supernatural effect
Blade guide is listed as (Su) in Tome of Battle, so what does that mean?
It cannot be counterspelled or dispelled
Supernatural abilities [...] are not subject to spell resistance, counterspells, or to being dispelled by dispel magic
So dispel magic and its kin do nothing to the blade guide.
It can be suppressed by antimagic field
Supernatural abilities [...] go away in an antimagic field
So antimagic field suppresses the blade guide (it disappears) if the blade guide enters the antimagic field, but it reappears 1d6 rounds after the antimagic field ends or the eternal blade moves away from it.
What disjunction does to supernatural abilities is an open question
Supernatural abilities are magical
Mordenkainen’s Disjunction [...] All magical effects [...] are disjoined.
So supernatural effects, being magical, are “disjoined.” But we have no idea what that means for a supernatural effect. The spell never mentions them specifically. It lists out several other forms of magic and what happens to those, but none of those things are categories that include supernatural effects.
For spells and spell-like ability, it means being dispelled as with dispel magic – but dispel magic does nothing to a Supernatural ability. So does disjunction just do nothing to supernatural abilities, as if it were just a super-dispel magic? That’s possible. Or does disjunction end a supernatural effect, as if it were dispelled and as if that were a thing that could happen to supernatural effects? Also possible!
Alternatively, disjunction destroys the magic on items, and can destroy antimagic fields and artifacts entirely. Does it perhaps do that to a Supernatural ability? Seems less likely (Supernatural abilities are far more similar to spells or spell-like abilities than they are to magic items or artifacts), but it’s still possible. Maybe being caught in a disjunction just eliminates all of your supernatural abilities wholesale! For once, if that happened, the blade guide would not just reappear 1d6 rounds later in the eternal blade’s space.
Or maybe a third, completely distinct thing happens to them. That’s possible too. Maybe the supernatural ability just enters a state of “being disjoined,” which has some weird quasi-philosophical ramifications.
The answer is, we just do not know; the rules did not specify what happens. Ask your DM.