Perhaps telling us what APK you are talking about... Even with Unknown Sources enabled, Google Security Verify Apps still does a verification of the APK for malware or known problems and sends anonymous information to them about the app. Not saying this is the issue, but it's possible. Also be aware that if the APK is broken or corrupted, or not compatible, it will not give you an Install button.
This. I had the same issue while trying to install APKs, once I paused the Twilight app it worked fine. The APK install screen doesn't warn you a screen overlay has been detected like most apps do.