Does Shakespeare have only heroines and no heroes?

First, we need to think about the relationship between the notion of protagonist and "hero" or "heroine." The notion of a hero belongs to oral tradition and orally derived works, and is normally a relatively flat character, usually of noble background, that excels in distinctly gendered virtues, warlike prowess for men and family loyalty and self-sacrifice for women. The epic hero follows an arc that normally culminates in triumph. 

Tragic protagonists differ from epic heroes...

First, we need to think about the relationship between the notion of protagonist and "hero" or "heroine." The notion of a hero belongs to oral tradition and orally derived works, and is normally a relatively flat character, usually of noble background, that excels in distinctly gendered virtues, warlike prowess for men and family loyalty and self-sacrifice for women. The epic hero follows an arc that normally culminates in triumph. 

Tragic protagonists differ from epic heroes in being more three dimensional and in following a plot arc that declines from good to bad fortune due to a combination of bad choices, character flaws, and fate. Shakespeare, though, is a modern dramatist, and his protagonists do not follow the epic model. Comedy as a genre does not have heroes.

Many of the characters in Shakespeare follow the pattern of the tragic hero. Both Macbeth and Lady Macbeth are strong aristocratic characters whose inherent flaws lead to their downfalls. Similarly, Othello, a brave and noble character whose jealousy, inflamed by the evil Iago, leads to his downfall, is a typical tragic hero. King Lear has heroic male and female characters as well as villains of both genders. Romeo and Juliet both follow similar narrative arcs of falling in love with each other and committing suicide. Thus if one uses the term "hero" as a literary term, meaning one conforming to the Aristotelian model of the tragic hero, Shakespeare's plays do have "heroes." Shakespeare's plays, not being epics, do not have epic heroes, and not being 21st century comic books or movies, do not have the comic book types of flat "good" superheroes who triumph over equally flat cartoon villains.