Boxers have pretty short coats. How does she do with weather in the 40's normally? or has she experienced it? There is one way to find out though, take her. ;-) I have become so used to hiking with my dogs (and camping), that I refuse to go without them anymore (one or the other). They are both Belgian Malinois, so their coat is slightly longer than your girl's, although mine are acclimated to fairly warm weather. That doesn't stop them from absolutely loving the snow.
My male is with me 24-7, 365 days a year. He is with me at home, when I'm sleeping, and when I'm at work (in fact he is my job). So if you couldn't guess, I'm a dog lover. I say start her off early and have many years of happy hiking with her. Learn some basic K9 first aid, and bring enough spare sleeping gear/ clothing so that if she gets too cold you can get her comfortable. But I believe most dogs are far more resillient to weather than we humans are. Also, 4 months is fairly young, so keep the distance manageable for her.
Just my .02 cents.
PS- Let's see some proud papa pics of her. I'm sure you have some. :)
(I'm not sure how to post pictures on this site or I'd tell you).