In the Material Design guidelines Google presented a new style of button, the FAB Button. I found instructions how to make it but I have trouble adding the shadow.
The problem with the circular shadow can be easily solved without any tricks with Outline: just add these properties to the button in the XML layout (in addition to the custom background):
android:elevation="5dp"
android:stateListAnimator="@null"
Although Android Studio may display it wrong in the layout preview, it works fine when launched on a device.