First of all the short story: You can't.
(This is due to the DVD-video being rerouted directly to your screen from the video-card (and bypasses the OS, rendering the standard command-shift-3 or 4 with a grayish-green block for keying in the video). Not restrictions from MPAA or whatever...)
Your friend's DVD-player is probably a strictly software-player.
Second: Even if you could get a screengrab of your 72 dpi screen in 300 dpi, all you would be left with is a miniscule image. (Or a scaled version). The resolution of any digital video is about 720x576 (for DV anyway), so anything higher would not leave you with higher res image.