Well, I poked around under there, couldn't seem to find anything. I pulled the swaybar off, went for a drive, and it still squeaks.
So I went inside, did a little research on ClubFrontier, and figured out the issue. It's the hood. When the hood is latched, either those little rubber bumpstops or the latch mechanism makes this little chirping sound. I can almost recreate it with my finger where the bumpstop rides on the front crossmember. I popped the hood and went for a (slow!) drive around the neighborhood, hitting every pothole I could. No more squeak.
Also, I'm not so sure about running without a front swaybar. There is definitely more front suspension movement, but there is also definitely more leaning. If I turn the wheel from 1/4 turn left to 1/4 turn right, I nearly feel like I'm going to fall out the window. That being said, under normal conditions, it shouldn't really be a problem. I just worry about emergency maneuvers. I'm going to drive with it a few days and see how I feel about it.