There's no denying that June Iparis is smart. In fact, she's the only known person in the Republic who got a perfect 1500 score on her Trial, and she's in college a whole lot earlier than other kids. June never loses her cool, and it's easy to see that she's going to be a great asset to the Republic and to the military.
When Metias—her guardian and brother—dies, June is utterly devastated, but she still manages to remain levelheaded when assessing the crime scene:
I don't even inch from the sting of her words. The details rush in, and I start talking. "Whoever hit him with this knife either stabbed him from close range or has an incredibly strong throwing arm. Right-handed." I run my fingers along the blood-caked handle. "Impressive aim. The knife is one of a pair, correct? See the pattern painted on the bottom of the blade? It cuts off abruptly." (1.4.63)
Being a prodigy makes it so that June rises up in the ranks a lot faster. She gets through college quickly, and she's allowed to join the military right after Metias's death. She even gets the high-profile mission of hunting down Day.
But being a prodigy can be a lonely life, and from the very beginning June's brilliance sets her apart from other people. She doesn't really have friends at school—she can't connect well with other people besides Metias—and no one understands how to associate with someone as smart as she is. That is, until she meets Day:
I can see her intelligence in every question she asks me and every observation she makes. But at the same time, there's an innocence that makes her completely different from most of the people I've met. She's not cynical or jaded. The streets haven't broken her. They've made her stronger instead.
Like me. (1.15.3-4)
Day may not know that June is a prodigy, and he doesn't even know that he got a 1500 on his Trial too, but he recognizes a kindred spirit. In Day June has found her match—someone who's just as smart, cunning, and street smart as she is. They make the perfect team because they understand each other fully and can keep up with each other.
June may display a great deal of patriotism and loyalty to the Republic at the beginning, but her main priority has always been her brother. Since their parents died when she was young, Metias has raised June and is the only family she has. In fact, he's the only real friend he has (besides Ollie the dog), so he's a very important person to her. When Metias dies, June decides that she must avenge his death because that's all she can think about—hurting the person who hurt her brother. She catches Day and thinks that this is going to help with her grieving process, but it just confuses her more.
"I really miss him," she whispers. "I thought he would be around for a long time, you know, someone I could always lean on. He was all I had left. And now he's gone, and I wish I knew why." (2.9.23)
Watching June suffer through the death of her brother and seeing her try to figure out why he died is a large part of the reason that Day has sympathy for her—he understands her grief and can see that she really cared about her brother. Instead of hating her for taking him into custody, he sees that she acted out of pain and out of love for her brother.
In the military of the Republic of America, it seems that a certain degree of ruthlessness is prized. After all, Commander Jameson is as cold as can be and doesn't give a whit about people's feelings, and Thomas is even worse. Thomas follows commands without thinking; he doesn't even care when he massacres a crowd, shoots Day's mom in the head, or even when he stabs his own friend Metias.
June, though, lacks the requisite ruthlessness and coldness to really cut it in the military:
I need to learn this, to familiarize myself with this. My ears ring from the spy's screams. I ignore the fact that the spy's hair is straight and dark like my own, and his skin is pale, and his youth reminds me of Metias over and over again. (1.10.28)
She tries to justify the cruelty and the torture tactics in her mind, but each subsequent act of violence makes her more and more uncomfortable. She gets upset when Thomas shoots Day's mom and is horrified when the military uses force against rioters. She can't turn a blind eye to the pain and injustice of it all; she sees all of the horrifying implications and personal tragedies that people are facing.
Because of this compassion, June cannot carry out her orders in good conscience. She ends up completely turning her back on the Republic and helping out Day because she can see that the military will hurt people without even thinking about it.