Help with questions about hardware-accelerated live streaming.

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I'm not ready to upgrade my motherboard and CPU yet as it does everything I need with the exception streaming games. I was thinking about buying a video capture box and offloading the encoding to my laptop as my tests show that it's powerful enough to handle just the encoding if I stick to 720p (though I’d like to do 1080 or even 1080+). However, I'd much prefer to keep everything localized on the same machine.

Is there some type of dedicated internal card or hardware solution designed to offload the stress of encoding a live stream in Twitch, YouTube, etc.? I've looked around and found some devices with mixed reviews for H.264, but it's unclear whether those devices are A: good and B: work with live streams.

I'm also a bit confused on whether they'd actually help with the encoding on Twitch and YouTube. As far as my feeble mind understands it, Flash video essentially acts as a wrapper around the video file similarly to .mkv and .mov files. While I know it’s not exactly apples to apples, what I’m trying to figure out is whether, assuming that is the case, I could encode in H.264 and essentially pass it through FMLE with minimal stress on the CPU.

Thanks in advance and any recommendations you might have for such hardware would be greatly appreciated. It must work in Windows 7, but Debian-flavored Linux compatibility would be a very nice bonus and OS X would be the cherry on top (but far lower down the priority list than the other two).

My Intended Setup (Bandwidth won’t be an issue.)
CPU: i7 930 OC'd 3.41GHz
RAM: 20GB DDR3 1600MHz
GPU: GTX460 1GB OC'd
Sound Card: Xonar Essence STX
Microphone: Blue Yeti Pro

Webcam: Microsoft LifeCam 
• Ideally I'd like to be able to use this with PiP and switch back and forth between this and the game stream occasionally.

Monitors: Two 1920x1200 monitors
• Only one is used while gaming and is usually run at native resolution, but I'd likely change to 16:9 for streaming.

Storage (Relevant drives):
• 64GB SSD for OS
• 2TB WD Black for Games
• 1TB WD Black for storage of videos being streamed
• 320GB drive which could act as a scratch disk if it'd help.

Intended Streaming Software
• Open Broadcaster Software (OBS)