I don’t believe it’s possible to persist the particular session, but you should be able to save the tokens you receive on a successful log-in via AsyncStorage and then fetch them when your app is launched so the user doesn’t have to enter the credentials every time.
As for making a button, you’ll want to make use of TouchableOpacity, TouchableHighlight or TouchableBounce to wrap your button component (Views and Text). You’ll then handle the signing out logic via the onPress property whatever Touchable you used.