No, anybody does not mean the same as any body or any corpse (but the latter two are roughly equivalent in this context). As noted by FumbleFingers in the comments, this is a mistake on Word's part, because automated grammar analysis is quite difficult to do well. Anybody is a synonym for anyone, and that's definitely not what you want to say here.
Given the context, I take it the speaker in the passage is searching the corpses for possible survivors, and wants to check every body that moves. I'd change the phrasing to this:
If you find any bodies with even the slightest movement
I suggest any bodies because...
- There are hundreds of bodies
- There may be more than one moving body
- Every body showing signs of movement merits an emergency call
If the speaker were searching for any body, then the search could stop after finding one, because any [single] body satisfies the condition (showing any amount of movement).
MrWonderful rightly points out that the use of any corpses sounds strange, because there's no question about whether a corpse is dead or not.
I would not omit bodies, because If you find any with even doesn't work well without some preceding dialogue specifying the ellipted noun. It begs the question any what; bodies, survivors, animals, rock slides? The context does make it clear to the reader, but it takes mental work to dig out. The listeners in the story might not find it so clear, especially since they're likely distracted and stressed by the tragedy at hand.
I've also switched the order of even, placing it after with, though this has nothing to do with the use of any bodies. Moving even changes the semantics of the clause. Even with emphasizes the bodies; the search becomes for bodies, even those which may be slightly moving, which includes unmoving corpses. But the speaker here is looking exclusively for bodies with [even the slightest] movement, meaning that any degree of motion greater than zero qualifies (so corpses are excluded). Both constructions convey that moving bodies will be rare, but their conditionally qualifying sets differ.