Can you get e-books from barnes and noble to your kindle even if you don't have a nook?

You can, but it probably won't work unless it's compatible.

The Kindle can read a great deal of formats, but it's preferable that you use their own: .azw format. It will read DRM-free .mobi (so public domain books are fine) and PDFs, etc., but don't purchase ebooks from other major online stores until you've made sure they're compatible.