So I figure I'll need an antenna, but I'm confused about what I'd need for DVR/PVR capabilities
There's a whole forum here at AVS that does nothing but talk about local HDTV reception, there's A LOT of antenna talk in there, best to read up on what that's like where you live, where the towers are, your line of sight to them, how far away they are, etc and then get a model antenna that's best for your area. You getting good OTA reception has little or nothing to do with which particular El Gato tuner you choose. Like geeji already said, both of the models you're considering can record high def over the air, so get whichever one suits your budget. (Me, personally, I wouldn't buy a USB stick tuner that relies on bus power and instead spend a little more to get the Homerun because overall it's a less problematic and more solid device. But, what you want to record is something called ATSC, so feel free to experiment with any tuner that lists ATSC in its specifications.)
am I correct in understanding that I then connect one of these to the Mac mini + antenna, and the EyeTV3 software then records programs to the Mac mini HDD?
You can also record to some other drive, like an external, beside the internal drive of the Mini.
Is the 2007 Mac Mini powerful enough to record and playback OTA HD signals?
Record, yes, playback possibly not in full screen HD (1920x1080p@60Hz). Depends on the CPU frequency and video card.
I'll be a little more specific than geeji here: even if your HDTV is an older 720p set and not a newer 1080p model, with either 2007 mini you won't get perfect playback of 1080i content because of two factors which you can't overcome: 1) neither the 1.83 nor the 2.0 C2D CPU is powerful enough to use EyeTV's best deinterlace setting, so you'll have to drop down to the second best setting, called motion adaptive, which noticeably compromises playback and 2) the GMA 950 was also terribly underpowered even for 2007, now it's a dinosaur.
But, since you're getting that mini for free, it wouldn't hurt to get started anyway, see how you like the setup, and that experience will help you decide whether upgrading to a more capable Mac HTPC is for you. Anything you record now with that Mini will be in perfect shape, recording ATSC isn't intensive at all. Once you decide you do like the notion of a Mac HTPC, though, your best bet may be to sell that Mini, and apply the proceeds toward something newer.