First of all, streaming at 1080p60 is a terrible idea. Flash is really bad at 1080p playback, and computers of even average power struggle with it. If viewers don't have the ability to change quality settings while watching your channel, then you'll get a lot of people having trouble watching.
If you still want a capture that can capture and output at 1080p60, though, then your two main consumer-level options are the XCAPTURE-1 and the SC-512N1-L, both from Micomsoft. The XCAPTURE-1 is a USB 3.0 device, and is very good, and can capture HDMI at 1080p60. However, you have to have an Intel or Renesas USB 3.0 controller for it to work. The SC-512N1-L is basically an internal version of the XCAPTURE-1, but instead of HDMI capture, it has DVI capture. You can easily get an adapter for HDMI -> DVI, but DVI does not support audio, so you'll have to find a way to split the audio off of the HDMI signal in order to capture it with the SC-512N1-L.
The Live Gamer HD can capture 1080p60 source video, but it only outputs at 1080p30. It also can downscale a 1080p60 source to 720p60 and output that, so that could work too.
Just don't stream at 1080p60, for your own sake and for the sake of your viewers.