How can I add a progress dialog when tapping a button in Flutter?

You can manage it using state as updating state rerender your widget and show content according to your condition . Kindly find below updated code and let us know if something is missed over here

Note :- I am assuming over here that your widget is extending StatefulWidget not StatelessWidget as you added build method only

Widget build(BuildContext context) {
    return Scaffold(
      backgroundColor: Colors.white,
      appBar: AppBar(
          title: Center(child: Text('Login')),
          backgroundColor: const Color(0xff2e80e4)
      ),
      body: _isLoading ? const Center(child: CircularProgressIndicator(),) : SingleChildScrollView(
        child: Column(
          children: [
            Padding(
              padding: const EdgeInsets.only(top: 60.0),
              child: Center(
                child: Container(
                  width: 200,
                  height: 150,
                  child: Image.asset('assets/logo.jpg'),
                ),
              ),
            ),
            Padding(
              padding: EdgeInsets.symmetric(horizontal: 15),
              child: TextField(
                decoration: InputDecoration(
                    border: OutlineInputBorder(),
                    labelText: 'Username',
                    hintText: 'Please enter your username'
                ),
              ),
            ),
            Padding(
              padding: const EdgeInsets.only(left: 15.0, right: 15.0, top: 15.0, bottom: 0),
              child: TextField(
                obscureText: true,
                decoration: InputDecoration(
                    border: OutlineInputBorder(),
                    labelText: 'Password',
                    hintText: 'Please enter your password'
                ),
              ),
            ),
            TextButton(
              onPressed: () async{
                await Navigator.pushNamed(context, '/forget_password');
              },
              child: Text(
                'Forget Password',
                style: TextStyle(color: const Color(0xff2e80e4), fontSize: 15.0),
              ),
            ),
            Container(
              height: 50.0,
              width: 250.0,
              decoration: BoxDecoration(
                  color: const Color(0xff2e80e4),
                  borderRadius: BorderRadius.circular(20.0)
              ),
              child: TextButton(
                onPressed: () async{
                  setState(() {
                    _isLoading = true;
                  });
                  //here I need to start the progress dialog
                  http.Response response = await postRequest();
                  Map<String, dynamic> parsedData = jsonDecode(response.body);
                  LoginResponse loginResponse = LoginResponse.fromJson(parsedData);
                  if(loginResponse.verificationStatus)
                  {
                    setState(() {
                      _isLoading = false;
                    });
                    await Navigator.pushNamed(context, '/home');
                  }
                  //Here I need to stop the progress dialog
                },
                child: Text(
                  'Login',
                  style: TextStyle(color: Colors.white, fontSize: 25.0),
                ),
              ),
            ),

            SizedBox(height: 100.0),
            TextButton(
              onPressed: () async{
                await Navigator.pushNamed(context, '/sign_up');
              },
              child: Text(
                'New User? Create Account',
                style: TextStyle(color: const Color(0xff2e80e4), fontSize: 15.0),
              ),
            )
          ],
        ),
      ),
    );
  }

As you mentioned that you are beginner , I would like to suggest to go through State management in flutter as it will help you to do many things

https://docs.flutter.dev/ui/interactive