She feels safe with him. Not that River Tam was weak by any stretch of the imagination, but sometimes it's nice to just be a woman instead of a weapon. Most days she thinks John Casey is the only person on the planet that realizes River is not just a tool to be used when necessary.
The Colonel is also the only person she feels comfortable around. Surprisingly, John Casey's mind is the most orderly that the reader has ever encountered. It's the most important factor in her requesting him as her handler after the Intersect's team rescued her—that, and the man's mastery of weapons. As much as River wants to be seen as a woman, she knows that she is also a very dangerous weapon.
Most of all, she trusts him. With all of her behavioral programming it would be easy for her handler to just take her free will. But she's seen into Casey's very soul, and knows that he would never betray her in such a fashion. Free will is the most basic and inalienable human right there is, and she knows he would sooner commit treason than take that right from her. Despite his reputation as a stone cold killer, River knows that her handler is a good man. A man that would do everything in his power to protect her and aide her in any missions that came her way. A man that she hopes to spend the rest of her life with.