My Droid X is running Verizon’s official build of Android 2.3.3 (Gingerbread) which is version 4.5.588. Luckily, there’s a very easy tool for rooting this: GingerBreak. Once you have root access you can run the BusyBox installer. I then used adb to get a shell and remount the /system filesystem read/write. I edited build.prop using vi to change two lines so the phone would pass Netflix’s model check. After that, the Netflix app just works. You do need the latest version and not the original leaked apk.
Update: Now with soundtrack!