The Conjure Animals spell says:
You summon fey spirits that take the form of beasts [...] each beast is also considered fey. [...] The DM has the creatures' statistics.
The fey spirit(s) summoned is a beast and it has all the statistics that the beast has as they appear in the monster manual. Note that most beasts don't actually have a language.
The Giant Eagle, though, is special. Its stat block does list a language that it knows and therefore can speak; the Giant Eagle language.
The Tongues spell says:
This spell grants the creature you touch the ability to understand any spoken language it hears. Moreover, when the target speaks, any creature that knows at least one language and can hear the target understands what it says.
Since the Giant Eagle has a listed language that it can speak, casting Tongues on it will grant you the ability to communicate with it as you wish.
Normally, casting Tongues on a Conjure[d] Animal would allow the animal to understand you but not allow you to understand it because most beasts are unable to speak.
According to page 9 of the monster manual:
Sometimes a monster can understand a language but can't speak it, and this is noted in its entry. A "-" indicates that a creature neither speaks nor understands any language.
Most creatures are summoned unwillingly, or are otherwise unhappy about being summoned.
The Monster Manual is rife with examples of creatures that hate being summoned, under almost any circumstances. For example:
Devils: MM (67):
However it is summoned, a devil brought to the Material Plane typically resents being pressed into service. However, the devil seizes every opportunity to corrupt its summoner so that the summoner's soul ends up in the Nine Hells. Only imps are truly content to be summoned, and they easily commit to serving a summoner as a familiar, but they still do their utmost to corrupt those who summon them.
Yugoloths (a specific circumstance) (MM 311):
A yugoloth summoned using its true name, as inscribed in the Books of Keeping, is forced to serve its summoner obediently. The yugoloth hates being controlled in this manner and isn't shy about making its displeasure known.
Elementals (MM 123):
Certain spells and magic items can conjure an elemental, summoning it from the Inner Planes to the Material Plane. Elementals instinctively resent being pulled from their native planes and bound into service. A creature that summons an elemental must assert force of will to control it.
Demons (MM 53):
A mortal who learns a demon's true name can use powerful summoning magic to call the demon from the Abyss and exercise some measure of control over it. However, most demons brought to the Material Plane in this manner do everything in their power to wreak havoc or sow discord and strife.
(MM 51):
If a single mistake is made, a demon that breaks free shows no mercy as it makes its summoner the first victim of its wrath.
One exception is a Glabrezu (MM 53):
A glabrezu takes great pleasure in destroying mortals through temptation, and these creatures are among the few demons to offer their service to creatures foolish enough to summon them.
The Gate spell is explicit about simply pulling any creature away without notice:
When you cast this spell, you can speak the name of a specific creature (a pseudonym, title, or nickname doesn't work). If that creature is on a plane other than the one you are on, the portal opens in the named creature's immediate vicinity and draws the creature through it to the nearest unoccupied space on your side of the portal. You gain no Special power over the creature, and it is free to act as the DM deems appropriate. It might leave, Attack you, or help you.
The Conjure Fey spell implies that the fey resents being summoned, since it attacks you if you lose concentration (much like a demon):
If your Concentration is broken, the fey creature doesn't disappear. Instead, you lose control of the fey creature, it becomes hostile toward you and your companions, and it might Attack.
Confusingly, the Conjure Woodland Beings spell does not have that clause. Indeed, you could summon the same creature with either spell, and it will only attack you when you lose concentration only if you used the higher level version.
The only case that might allow for some free will is Conjure Celestial, because it has the freedom to only obey commands that follow its alignment.
It obeys any verbal commands that you issue to it (no action required by you), as long as they don't violate its alignment.
Best Answer
If the spell required you to share a language with the creature, it would say so.
Let's look at something like Modify Memory (PHB, page 261)
It's specifically called out that the target must be able to understand your language.
Command (PHB, page 223) is another example:
Since there is no mention of language in Conjure X's description, it will follow your commands regardless of the language you speak them in (although I think it would be a reasonable assumption that it has to be a known language within the DnD universe, not just one you made up on the spot).
Note though that the spell only says
It doesn't allow you to automatically understand the creature's language, give the creature the ability to actually understand your language (unlike the Find Steed spell), or give it any special ability to communicate with you.
As per @SevenSidedDie's comment, they may not even understand the actual commands, being driven more so by magical impulse at the intent behind your words rather than by the words themselves.