'If, then, they say to you, "Look, he is in the desert", do not go there; "Look, he is in some hiding place", do not believe it; because the coming of the Son of Man will be like lightning striking in the east and flashing far into the west.  Wherever the corpse is, there will the eagles gather.'.  Matthew 24:26-28

How a verb helps to reveal the meaning behind the corpse and the eagles in Matthew 24:

Gather (verb) - collect; round up; pick up.

Gathering (noun) - people assembled; congregation; get together; meeting; rally.

Oxford mini reference dictionary and thesaurus.

If we substitute 'gather' for 'pick up' and read the verse again,

Wherever the corpse is, there will the eagles pick up,

instead of picturing a gathering of eagles we get a different understanding of what the Messiah is saying in Matthew 24:28.

The reason why this is so important is because the Messiah is giving practical advice to the remnant of Israel in the end times so that they can be comforted that there is an escape from the beast and his army who are coming to kill them.

In Exodus 19:4 Yahweh says to His people, Israel, through His prophet, Moses,

"You yourselves have seen what I did with the Egyptians, how I carried you on eagle's wings and brought you to myself.". 

The phrase 'eagle's wings' is symbolic of God's help and in Matthew 24:28 the eagles are also symbolic of God's help.  

In Isaiah it says, 

"Even youths shall faint and be weary and young men shall fall exhausted; but they who wait for the Lord shall renew their strength, they shall mount up with wings like eagles; they shall run and not be weary; they shall walk and not faint.".  Isaiah 40:27-31.

In Revelation it says,

"As soon as the devil found himself thrown down to the earth, he sprang in pursuit of the woman, the mother of the male child, but she was given a huge pair of eagle's wings to fly away from the serpent into the desert, to the place where she was to be looked after for a year and twice a year and half a year."  Revelation 12:13-14. 

The woman is Israel and the male child is Yeshua Ha'Mashiach.

In Exodus, Isaiah and Revelation the eagles and eagles wings are symbolic of God's help, therefore, they do not represent either the Roman or U.S. armies.

If the eagles are God's help then what is the corpse?  

A corpse is a body and Christ's followers are the body of Christ.  In Matthew 24:28 the remnant of Israel are the corpse and help will arrive from God to gather (collect) them wherever they are and assist them to flee from the beast army.

'Wherever the corpse is, there will the eagles gather.'

In Matthew 24:26  Jesus warns the remnant of Israel in the end times not to listen to others who will try to get them to go into the desert or some hiding place to look for the Messiah instead they are to wait for God to bring them to himself.

'If, then, they say to you, "Look, he is in the desert", do not go there; "Look, he is in some hiding place", do not believe it;

In Isaiah there is also mention of waiting for the Lord's help,

"Even youths shall faint and be weary and young men shall fall exhausted; but they who wait for the Lord shall renew their strength, they shall mount up with wings like eagles; they shall run and not be weary; they shall walk and not faint.".  Isaiah 40:27-31.

God's help will arrive very quickly 'like lightning striking in the east and flashing far into the west.'.  There will be no time to prepare a journey and go looking for the Messiah.  

For the remnant of Israel in the end times it is a matter of faith that Yahweh will not abandon His people to the beast army; the Lord himself will appear wherever they wait and gather them together.  They will follow Him to a place of safety prepared for them for three and a half years where Yeshua Ha'Mashiach will personally teach and prepare them to live in Jerusalem in His millennial kingdom.  Praise Yahweh!