Captain Miller reveals his background to the squad he's a school teacher, and uses it to explain why he wants to complete the mission to save Ryan.
Miller: "So, I guess I've changed some. Sometimes I wonder if I've changed so much my wife is even going to recognize me, whenever it is that I get back to her. And how I'll ever be able to tell her about days like today. Ah, Ryan. I don't know anything about Ryan. I don't care. The man means nothing to me. It's just a name. But if... You know if going to Rumelle and finding him so that he can go home... If that earns me the right to get back to my wife, then... that's my mission."
Jackson shoots the sniper right at his eye, a moment after the sniper spotted Jackson aiming at him.
The cinematography of the battle sequences are astonishing; not only do they capture the chaos and brutality of the fighting, but a lot of complicated sequences are shot in long, single takes, a perfect orchestration of cast and crew making something immersive come to life.
The eponymous Ryan, when encountered, is expected to be some detached, delusional rich and/or momma's boy wanting to play soldier. When the squad finally meets him face to face, they find anything but. Let's go through the list, shall we?
At first, he's confused by the order to bring him home, even upon learning about the deaths of his brothers—all of whom it should be added, he GREW UP WITH. And yes, it's very clear that he's conflicted and broken up by the news. You would be, too.
Second, he immediately begins bringing up everything he can muster to mind in regards to military protocol—not to aid his escape from the war, but to prevent it.
Third, and one of the hard-hitting moments: his tirade about how he shouldn't have to leave this whole mess is interrupted by Reiben (one of the squad sent to retrieve him) stating that two of their men have already died trying simply to FIND him...upon which he doesn't sulk, nor lash out. He simply turns away from Captain Miller to ask what their names where—even Miller at this point realizes he's become a third wheel in the conversation, and it shows.
Fourth, upon hearing those names, and fully processing everything being told to him, Ryan walks away from everyone present, and just breathlessly utters to no one in particular, "It doesn't make sense." He then proceeds to rant about the whole mess; he points out the other present members of his unit, stating that every single one of them has fought as hard as he has, and by proxy deserves this chance to go home and wash hands as much as he does. The film gives brief, but poignant shots of three or four other members of the unit, all whom are silent, but their emotions clearly mixed: on the one hand, they all clearly think Ryan deserves this reprieve on some level. On the other, they all clearly think they do, too. That speaks both for Ryan himself as a soldier and friend, and for the situation in general.
Fifth, after all of that, Miller takes the one card he has left: appealing to Ryan's sense of family. "Is that what they're supposed to tell your mother? When they send her another folded American flag?" To which, after only one second or less, Ryan flatly and firmly states the following:
"Tell her that when you found me, I was here. And I was with the only brothers I had left. And there was NO WAY I was going to desert them. I think she'll understand that."