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Yes, they can charge. They can also run, withdraw, and 5 ft step, etc...

What are you talking about here, HOuse RUles?

By estrict rules interpretation, a creature cannot RUN with the FLying Skill, nor can it run with the FLY spell by RAW. What I mean, is that this a normal assumption situation, that usually arises with the complex uasage of 3-d positioning anf of course FLY-oriented ruling.

According to R.A.W., a creature with with the FLY spell cannot run, and this extricly violates d20 ruling and makes your answer seriously invalid StreamoftheSky...even if you mean this could be accomplished soehow, you are being blinded by the Flying rules interpretation, and FAQ/errata clarifications as per today SRD flying rules 2014 & here too: PAIZO´s flying rules 2014

Can a creature with a flight speed, an eagle in this case, charge and get the +2 bonus on attack?

Or is charge strictly land-bound?

Let me put it top you this way Quatar: What would be the DC, for the Flying creature´s Maneuver, when charging? Quick answer, none, because ¨CHARGING¨ violates the rules of Pathfinder two-dimensional movement. Look at Sidebar

There are few example abilities that actually supersede this ruling, call then exceptions if you will, and some should be mentioned here--not limited to the folowing:

- Fly Spell, it indicates a creatre extricly affected by this spell may charge, but not run FLy spell

- Snalligaster´s Aerial CHarge, and Ex Ability unique to this critter Bestiary 4´s Snalligaster

For clarification, Note that comming to a full stop is also impossible for most manuaverability classes of flight in D&D without stalling out. Just look at what you could do with proper usage of the flying skill here quick flying rules

Conclusevely, even if the CHARGE action per-se shows no limitation -or restriction on wheter it can be used according to one´s speed, unless your character actually has PERFECT maneurability (granted by the FLY spell), it is a violation of the rules, allowing a Flyng creature to CHarge on this manner.

Please note that such estrict interprertation, actually limits game mechanics that benefit from Charging (if not used by a creature affected by the FLy spell), such as (but not limited to) the following:

- Spirited Charge
- Pounce
- Mounted Skirmisher
- Charging Hurler

Similarly, there are abilities that would permit similar movement, wihout vilating these ruling, such as (but not limited to) the following:

- Dive: It is essentially a charge action (with the +2 to hit and -2 AC) that inflicts double damage with a claw attack. dive is an action that flying creatures can take, detailed in the MM, a SU or Ex depending on the creature.

- Flyby attack: It is a feat that allows a flying creature to make a partial action at any time during its move (rather than only before or after moving).

- Death from Above(Combat Feat) You allow gravity to add extra force to your charges. Benefit: Whenever you charge an opponent from higher ground, or from above while flying, you gain a +5 bonus on attack rolls in place of the bonuses from charging and being on higher ground.

- Mythic Death from Above: This is hardcore as hell. If you hit at all, you get a free action trip attempt, and if you score a critical hit, the crit modifier of your weapon in increased by 2.

You can entertain some powerplay if...Say you can have the creature hit there..just combine Punishing Kick and Vicious Stomp for the same flavor effect. You dropkick someone into the ground, then then stomp on them as they're prone. Or similarly use Shield Slam, instead.

Further clarification can be found here : Dive-and-Flyby-Attack @ EN world