Fist of all I get that the original purpose of the Aegis is more of a tank, to soak some damage and return the favor. But as I get deeper into Path of War, I see that they can use some of its abilities. Are any of the ranged stance, boosts, strikes, or counters applicable to the Aegis' range attacks (Ranged Attack or Energy Blast)?
Both of these customizations can potentially benefit from boosts and stances. They are both ranged attacks, but not weapon attacks, so you have to carefully read each boost and stance to see if it is compatible. They also lack any ammunition, and are not thrown. Many boosts and stances will specify melee attacks, weapon attacks, ammunition, and/or throwing, and so not work with these customizations. Note that you have to check the full description of the boosts and stances to be sure—the abbreviated descriptions in the list are, well, abbreviated, and may leave out caveats like “melee” or “weapon.”
The ranged attack customization can also be used to make attacks called for during a strike. Again, it is a ranged attack and not a weapon attack and so that limits which strikes it is actually compatible with.
The energy blast customization cannot be used to make attacks during a strike: it is its own standard action, and you cannot take a standard action in the middle of some other action used to make the strike. Even if you somehow had two standard actions in the same turn, you would have to finish the strike (with some other attack) first, and only then use energy blast.
However, note that some maneuvers have inconsistent wordings, depending on where you read them. For instance, the Stance of Piercing Ray is applied to ranged attack in its description, but the summary at the top of the page talk only about "fired ranged weapons / thrown weapons", and the fluff says "[you] cause [your] ammunition to pierce like the burning rays of the sun itself".