I just checked and mine defaults to HQ when launched from the browser and the option to change is there.
Not sure what the deal is
I'm not sure this is worded incorrectly or if I'm misinterpreting it. When you said "launched from the browser", that makes me think you mean the YouTube app since it prompts you to either open the video in the browser or the YouTube app and you're choosing the YouTube app, which will obviously give you an HQ option.
If you're actually playing the video in a browser, it does not give you any video quality options. It does behave a bit differently between browsers, but none of them have quality options.
The stock browser, when in full screen mode, it just gives you the "<< ll >>" (rewind, play/pause, fast forward) buttons and has no HQ button/options.
In the Dolphin HD browser (only other browser I use), it actually looks like a normal YouTube video, even in full screen mode with all the normal video options (play/pause, progress bar, full screen button) except quality options.
At least this is how it is on my phone running NexBeast v0.1.1 (FRF57). I'm running the 3rd (newest) beta release of Flash right now too.