You can't use most apps without also using Google firmware, since most developers use Google firmware calls, rather than writing code that's already been written. So you can use Maps, but you're also using Android OS, Android System (both of which are Google software) and Google Services. You don't have to use anything else, but a lot of things will be running in the background in any Google device (after all, "Android", which is a Java runtime, runs on Linux, which is also running in the device).
You could use MapQuest GPS Navigation & Maps, if you're so anti-Google, but if it's an Android device, you can't "not run Google software" completely. If you just want a map program, buy a GPS device, not an Android phone.