I birthed in what is called in the Bradley book a sitting squat (Natural Childbirth the Bradley Way..excellent book, please consider getting it, it has great advice on how to achieve a natural birth in a hospital). It is "actually a full squat, except that you are tipped back on your bottom to make it a sitting squat." There are pictures of it in the book. The hospital I was in didn't mind it at all, probably because I was in the bed and it looks similar to their usual lying position. I held back my own legs, and pushed on my own (self directed pushing) without someone telling me. I also declined the IV and monitor belt. You just have to be brave and firm about what you want.
Ultimately, how well your desires/requests are taken and implemented depends on who your provider is. Ideally, you'll be able to talk to them ahead of time and they'll know how you feel and respect it. If you're not as familiar with who's on call when you go in, then once again, just be polite, but firm about your requests! You'll be happy you did when the birth of over and you were able to achieve to some degree the birth experience you wanted. I say to some degree because birth is unpredictable, there will be something that won't go as planned, but don't worry about that. Just do the best you can!
One other small tip....if you feel pressure from folks around you or feel overwhelmed and need some privacy......ask to go to the bathroom, whether you need to or not (hopefully there will be a bathroom in your birthing room) and stay in there as long as you like. Lock the door! My sil did it and it really energized her. She came out when she was good and ready and then proceeded to have a great birth!