Hi, I'm genderqueer and I want to start by saying that you must have a great relationship with your child and you should be proud of that even with all your concerns. I'm well into adulthood and have never told my parents (telling them about my sexuality was a big enough disaster...).
Trying to answer what's already come up: While some do find it later in life, my experience within the communities has been that the vast majority of trans* people know at an early age, we just don't have the words for it for a while as it's not really talked about. Some people do try on various identities at first trying to find something that fits and your child may find something else that fits better as she's obviously starting to explore and feels at this time this fits best so the firm starting point you already have of loving no matter what and not pushing any identity as important. If the child pushes forward that it's an important part of how they view themselves, that's fine, but one of the worst things for me was having my mother try to push me into having my sexuality as my main identity and giving me and judging things based solely on that - basically led to not being able to discuss anything with her.
Even if it is peer influenced, it's still her that is working through her identity and the opening up and playing with gender and gender expression within a peer group is far safer. I find it's very often described negatively and I don't think it should be - it's one of the safest ways to explore ones identity and much better than having these confusing feelings without the words or friends to discuss and explore our identity and options with. Feeling alone, lost, and wrong is not what we want and if someone feels they are genderqueer for a while but end up feeling they really they are cis but enjoy being both masculine and feminine, I don't see the harm or pressure in it. You can't make someone trans* anymore than you could make someone gay, and sometimes the first to announce will be followed by others both exploring and those that have known for ages but have been afraid.
The breast binder thing is concerning though - I would discuss what she thinks breast binders are going to do for her and have her look up the risks that come it and come up with ground rules. Basically, wearing them for too long periods can cause damaging to the muscles underneath - they're meant as short-term measures only and I'm worried that starting so young would mean the bigger invasive measures would have to come in soon due to damage caused. Never wear them to sleep in, don't wear them daily, and really I would recommend at her age starting with very snug sports bras and larger shirt, thicker when the weather allows it. I found lovely shirts as a teen really helped when I needed to look as masculine as I felt and wanted to register as male with those around me. I've personally never worn a binder, I find the right clothing choice works for me, but I have a smaller than average chest to start with so it's easier for me than for my larger assigned female at birth friends.
If you have other questions, I'll check back here and in my inbox. If you think she'd like a resource to help explain herself or explore ideas and terms, the Genderbread from It's pronounced Metrosexual is a great, neutral, friendly place to start - there are a few versions, you could print some or just show her the link after you've had a look at it.