I'm planning to rivet the trunions & rest of my receiver together tonight. After this I'll be using a plain vertical press to install my barrel. Do I need to do anything special or take any precautions to keep from bowing or buckling the receiver body while pressing the barrel in?
Yep. Don't brace against the receiver while pressing, only brace against the trunnion. The stamped receiver WILL bend if you use it for support while pressing a barrel.
You should use some type of assembly lube to minimize friction during pressing. A brass bolt should be slid loosely into the muzzle end to prevent damage to the crown (a penny or nickel works for this too).
You can work from the opposite direction, too. You can fabricate a split plate with a hole for the barrel and this would sit in the press. You would have to make an extension for the press ram that only makes contact with the trunnion. You would then position the receiver on the barrel and position the extension against the trunnion. Crank the press handle and start working the receiver down onto the barrel. This method is more stable and less likely to damage any parts. The downside is that the barrel pretty much has to be stripped. The sight blocks and gas block have to be removed. As a minimum, the Rear Sight Block should be pressed towards the front of the barrel to allow the split plate to make contact evenly with the shoulder of the barrel. If you make the plate right, you can use the different diameter landings on the barrel for support instead of the barrel shoulder. This would allow you to leave the RSB in place.
A builder named Xebec makes a kit with the split plate and ram extension. If you look in the photo section of Gunsnet, you can see his album.
This is the ram adapter for the trunnion: