Question

meaning of the phrase" looked him squarely in the eye"

Answer 1

here is the answer
Squarely" is considered more correct (it should agree with "look" and so should be an adverb), but I'd note that Americans at least definitely say it both ways. 

And yes, in this context, it means evenly, directly, or right on. The idea is that the subject is not avoiding your gaze, but confronting it.

Examples:
I looked him squarely in the eyes and told him that I did not like his tone of voice.
She looked me squarely in the eyes and told me she did not love me.
My son looked me squarely in the eyes and admitted that he broke the lamp.

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