Kylo Ren:

He has an affinity to cry to the music of The Lord of the Rings: The Return of the King, especially the scene where he cries with the hobbits at the ending of the film. When this scene plays, he cries softly in your arms, and this breaks your heart to see Kylo cry to this film.

general Hux:

He has told you many times, when you are working on the bridge, and it was his turn to take a break in your quarters, and he has a good cry to Baby Mine from Dumbo. He cries at this film every time he watches it.

Anakin Skywalker:

Yes, he likes a film that's soundtrack makes him cry too, he's fond of The fault in our stars. The soundtrack for it sets him off, and that's when your instincts to comfort him come in handy.

Luke Skywalker:

He likes to watch a movie with a heartbreaking soundtrack that brings tears to his eyes, his favourite character in Harry Potter got killed off, dumbledore so his heartbreak comes from the 6th Harry Potter movie.

Han Solo:

His film to cry to is Old Yeller, he always cries at the shooting of Old Yeller due to being rabid and the emotional ending scene. The soundtrack as well.

Obi-Wan Kenobi:

Like Luke, Obi-Wan cries at Harry Potter, but in film 7 part 1, his favourite character, Dobby the House Elf was killed by Bellatrix LeStrange's dagger as he helped Harry and his friends escape from Malfoy Manor. He cries inconsolably at Dobby dying in Harry's arms.

Finn:

He seems to like the song, Wind Beneath My Wings by Bette Midler, so the movie soundtrack he cries to is from Beaches, he especially cried at the main song. He cries so much that he called asleep after the movie ended.

Poe Dameron:

He cries at Twilight Saga: Breaking Dawn part 2, die to the one song by Christina Perri, A Thousand Years, and he always cries at the end of the movie, due to the song itself.