This is one of those cases of bad wording leaving it open to interpretation. Here's the wording for the Veils ability (emphasis mine):
Veils: When an initiate creates a warding, she can choose and imbue
the warding with any one veil she knows how to create. These veils
duplicate the layers of a prismatic wall and are described below. An
initiate's caster level for these veils is equal to her arcane
Warding is a Spell-like ability, which means it's vulnerable to being hit by Disjunction. Prismatic Wall says explicitly that the whole thing is removed by Disjunction:
A rod of cancellation or a mage’s disjunction spell destroys a
So, what does "duplicate" mean? Does it mean that it inherits Prismatic Wall's weakness? Maybe? Typically when they mean that, they use this wording (from Prismatic Sphere):
This spell functions like Prismatic Wall, except
Did they not use that wording deliberately to make this ability function differently, or was it an oversight? Nobody knows, and it's not addressed by the errata. Because it doesn't actually say that it's vulnerable to Disjunction as Prismatic Wall does and it doesn't "function like Prismatic Wall", does that make it immune?
An added thing here is that even if the Indigo Veil "blocked" it, what makes the Indigo Veil itself immune to it? Nothing in the description says the Veil itself can't be disjoined, and it has another explicit dispel condition (Daylight). Once the veil is down nothing protects the effects behind it anymore.
As for the Violet veil, destroying effects as if they were disintegrated has no effect on Disjunction. Disjunction's wording is clear:
All magical effects and magic items within the radius of the spell,
except for those that you carry or touch, are disjoined. That is,
spells and spell-like effects are separated into their individual
components (ending the effect as a dispel magic spell does)
Warding is a spell like ability, so without any wording that it's immune to disjunction I would say it's not (as disintegrate doesn't do anything to disjunction).
Based on all that (and my belief in what was intended here), my ruling as a GM would be that these two veils do not block Disjunction. That said, due to the confused wording here people could (and have) ruled otherwise in their games.
How many Disjunctions do I have to cast?
One. Disjunction is an area-effect magic remover (similar to the area effect version of Dispel Magic, but much more powerful). It doesn't get stopped by removing an effect, it keeps going and removes any effects behind that effect. The only exception would be if an effect blocked Disjunction itself (such as an Antimagic Field that it fails to remove).
If it removes the Indigo Veil (and this answer makes a case for why it should), then it removes every other veil at the same time, and every magic effect behind the veils that is in range, as per the normal effect of the Disjunction spell.
You only need to use multiple spells to take down the veils (or a Prismatic Wall) if you're using the color specific dispelilng spells. Disjunction hits all of them at the same time.
Note: Just in case there's any confusion on this point, Mordenkainen's Disjunction and Mage's Disjunction are the same spell. It has two names, depending on if you're looking in a published book or the SRD website.