Even if I'm personally working under linux, it is quite possible to develop for bitbox under windows.
The following article / page is a reference for it, and will be updated accordingly. Also, if there is any part missing, please tell it !
A very thorough reference can be found by example there : http://www.jann.cc/2013/10/10/embedded_development_with_open_source_tools_on_windows.html#install-the-gcc-arm-embedded-toolchain
Most of the current guide is taken from there.
It targets development on a cortex-m4 chip so it's quite similar and adds a ton of optional software which can be very useful. The proposed method is quite barebones (or light), because, well I tend to prefer lighter solutions, but integration with eclipse could be done I'm sure.
Alternatively, DIYDSP has an page with instructions to setup a linux virtual machine suitable for stm32 development, as well as various other tips.