Letterbox is a 4:3 picture with black bars matted on the top and bottom of the frame.
Squeeze is a true 16:9 picture with no bars. You lose a little bit of resolution in Squeeze mode, but it's not enough to really worry about. I shoot Squeeze most of the time.
When you bring it into your NLE, make sure you put it on a 16:9 timeline. Also when it comes time to render it out, select 16:9 then as well. You should be good to go.
If you crop the black bars off of the letterbox footage and throw it on a 16:9 timeline, it'll stretch it out to 16:9 and probably give you something similar to what shooting squeeze gives you.