You are effectively altering the pixel size when you squeeze everything in the timeline. Everything looks squeezed when using square pixels.
1920 x 1080 HD is square pixels always. There are no anamorphic presets, which is why I think they intend to stretch it back out in the projector.
There are HD formats that use anamorphic pixels (DVC PRO HD, HDV), but they are not 1920 x 1080.
I have received squeezed stuff for editing many times, and when I unsqueeze it in the edit, it appears in the proper aspect ratio (letterboxed) in a 16:9 frame. I have not been asked to deliver that way before, which is why I suggested you send them a test. Best to be sure before giving them the whole film.