Source, please? (For the claim that men are more afraid of (or actually are) being judged by other men than by women? Note that the scenario in question might be limited to friends but it's not limited just to family or BFF.)

The Man Box study doesn't quite address the issue, but it's not terribly consistent with your claim (because men in the box endorse stereotypical hegemonic masculinity but feel more comfortable crying in front of other men).