Do real Jewish men cry?
If we cursorily examine Tanakh, the answer comes back loud and clear: yes.
The verb בכה, cry, appears 116 times in Tanakh. Avraham, Yaakov, Esav, Yosef, the Egyptian people, the Jewish people, David, Yehonatan, Elisha, Chizkiyahu, Yirmiyahu, Ezra, and Nechemia all cry.
These cries represent a range of emotions, including crying upon emotional outbursts and crying as part of formal mourning rites; crying due to sadness and crying due to joy; crying for obvious reasons and crying for reasons unknown.
And while there are a few cases in which crying is criticized, particularly in the case of the Jews' crying in the desert for food, in most cases no judgment is passed.
Indeed, the case of the Jews in the desert aside, to the best of my knowledge crying is never deprecated throughout Tanakh. And in the cases of men such as Esav, Yaakov, David, Ezra, and Nechemia, it coexists quite comfortably with hunting, physical strength, war, and leadership.
Many of the greatest men in Tanakh were fierce warriors and leaders who were unashamed to shed tears.
עת לבכות ועת ושחוק; עת מלחמה ועת שלום