The below code might help you

Checkbox Group:

 var hobby= [];
 $.each($("input[name='Hobby']:checked"), function(){
     hobby.push($(this).val());
  });
  console.log("My hobbies are: " + hobby.join(", "));

Radio Button Group

var gender = $("input[name='Gender']:checked").val();
console.log("Your are a - " + gender);

Your empObj as below.

var hobby= []; 
$.each($("input[name='Hobby']:checked"), function(){ 
    hobby.push($(this).val()); 
}); 

var empObj = 
{ 
 CustomerId: $('#CustomerId').val(), 
 Name: $('#Name').val(), 
 Country: $('#Country').val(), 
 Gender: $("input[name='Gender']:checked").val();, 
 Hobby: hobby, 
};

In case of check boxes there can be multiple checked by user so you will need to get all those like:

$("input[name='Hobby']:checked").each(function(){
     console.log($(this).val());
});

This might also work for you but not sure:

var checkedValue = $("input[name='Hobby']:checked").val();

1. Code for Redio Button Or Check Box as following, i also use partial view

               <div class="form-group">
                   <label for="Gender">Gender</label><br />
                   Male
                   <input id="Male" name="Gender" type="radio" value="Male" />
                   Female
                   <input id="Female" name="Gender" type="radio" value="Female" />
               </div>
               <div class="form-group">
                   <label for="Hobby">Hobby</label></br>
                   <input type="checkbox" id="SHobby" name="Hobby" value="Singing">
                   <label for="Hobby">Singing</label>
                   <input type="checkbox" id="DHobby" name="Hobby" value="Dancing">
                   <label for="Hobby">Dancing</label>
                   <input type="checkbox" id="WHobby" name="Hobby" value="Writing">
                   <label for="Hobby">Writing</label>
   

i use add Function code as following , when i add data it take by Default Male for Gender and Dancing For Hobby

function Add() {
        debugger;
        }
        var empObj = {
            CustomerId: $('#CustomerId').val(),
            Name: $('#Name').val(),
            Country: $('#Country').val(),
            Gender: $('#Male').val(),
            Gender: $('#Female').val(),
            Hobby: $('#SHobby').val(),
            Hobby: $('#WHobby').val(),
            Hobby: $('#DHobby').val()

        };
        $.ajax({
            url: "/Home/Add",
            data: JSON.stringify(empObj),
            type: "POST",
            contentType: "application/json;charset=utf-8",
            dataType: "json",
            success: function (result) {
                debugger;


                loadData();
                $('#myModal').modal('hide');
            },
            error: function (errormessage) {
                alert(errormessage.responseText);
            }
        });
    }
    $(document).ready(function () {
        loadData();
    });

please help me to get proper selection value for Gender OR Hobby