Flutter : How to keep user logged in and make logout

You can make an entry in Shared preference after getting response code 200 from api.

 SharedPreferences prefs = await SharedPreferences.getInstance();
 prefs?.setBool("isLoggedIn", true);

then you can navigate user after checking status from shared preference

Future<void> main() async {
  SharedPreferences prefs = await SharedPreferences.getInstance();
  var status = prefs.getBool('isLoggedIn') ?? false;
  print(status);
  runApp(MaterialApp(home: status == true ? Login() : Home()));
}

Update :-

Another way of doing it is you can also add your logic into splash screen and splash screen should be entry point in your app

class SplashScreen extends StatefulWidget {
  @override
  State<StatefulWidget> createState() {
    // TODO: implement createState
    return _SplashScreenState();
  }
}

class _SplashScreenState extends State<SplashScreen> {
  @override
  void initState() {
    // TODO: implement initState
    super.initState();
    startTimer();
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Container(
        decoration: BoxDecoration(
          image: DecorationImage(
              image: AssetImage("assets/images/clinician_splash.png"),
              fit: BoxFit.cover),
        ),
      ),
    );
  }

  void startTimer() {
    Timer(Duration(seconds: 3), () {
      navigateUser(); //It will redirect  after 3 seconds
    });
  }

  void navigateUser() async{
    SharedPreferences prefs = await SharedPreferences.getInstance();
    var status = prefs.getBool('isLoggedIn') ?? false;
    print(status);
    if (status) {
      Navigation.pushReplacement(context, "/Home");
    } else {
      Navigation.pushReplacement(context, "/Login");
    }
  }
}

For logout add below functionality in onPress event of logout button :

void logoutUser(){
SharedPreferences prefs = await SharedPreferences.getInstance();
prefs?.clear() 
 Navigator.pushAndRemoveUntil(
      context, 
      ModalRoute.withName("/SplashScreen"), 
     ModalRoute.withName("/Home")
    );
}

For security :-

Here in example I have used SharedPreferences which is not secure.for security you can change SharedPreferences to flutter_secure_storage.

https://pub.dev/packages/flutter_secure_storage#-readme-tab-