Having worked at the airport (off and on) for the last four years (both day and night shift), I can tell you with certainty that no one will try to boot you out. For one, the airport is busier between midnight and 2am than it is any other time of day (we have tons of red-eye flights here, since it tends to work out better for connecting flights in the Lower 48), and so the downtime at the airport really only lasts from 2:30 to about 4:30. Second, you'll have lots of company, as two thirds of the benches around the airport (both inside and outside of security) are typically occupied with people doing the same as you, so stake out your bench early.

Concourse B is a little dimmer and may be easier to sleep in than the newer, brighter-lit Concourse C. However, the place I would pick would be the viewing area/Native art gallery located right behind security at the juncture of B and C concourses (the stairs/elevator up to it are behind the Starbucks that is opposite the Chili's Too). There are some benches up there and it's relatively secluded; plus, you can see the whole airport and watch planes if you're an airplane fan.

If you get really bored, come down to the [brand new] car rental center and say hi!