yogeshc says:
    [DisplayName("Email Id")]
    [Required(ErrorMessage = "Email ID is Required")]
    [RegularExpression("^([0-9a-zA-Z]([-\\.\\w]*[0-9a-zA-Z])*@([0-9a-zA-Z][-\\w]*[0-9a-zA-Z]\\.)+[a-zA-Z]{2,9})$", ErrorMessage ="Invalid Email Id")]
    public string email { get; set; }
You had defined required for email field. But email field is blank on post action. So you need to ignore the email field from ModelState.
ModelState.Remove("email");
Refer below sample.
Model
using System.ComponentModel;
using System.ComponentModel.DataAnnotations;
public class user
{
    public int id { get; set; }
    public string name { get; set; }
    [DisplayName("Email Id")]
    [Required(ErrorMessage = "Email ID is Required")]
    [RegularExpression("^([0-9a-zA-Z]([-\\.\\w]*[0-9a-zA-Z])*@([0-9a-zA-Z][-\\w]*[0-9a-zA-Z]\\.)+[a-zA-Z]{2,9})$", ErrorMessage = "Invalid Email Id")]
    public string email { get; set; }
    [DisplayName("Username")]
    [Required(ErrorMessage = "Username is Required")]
    public string username { get; set; }
    [DisplayName("Password")]
    [Required(ErrorMessage = "Password is Required")]
    [DataType(DataType.Password)]
    public string password { get; set; }
    [DisplayName("Remember Me")]
    public bool RememberMe { get; set; }
}
Controller
public class HomeController : Controller
{
    // GET: Home
    public ActionResult Index()
    {
        return View();
    }
    [HttpPost]
    public ActionResult Index(user u)
    {
        ModelState.Remove("email");
        HttpCookie cookie = new HttpCookie("User");
        if (ModelState.IsValid)
        {
            // Rest of code.
        }
        return View();
    }
}
View
@model ModelState_False_MVC.Models.user
@{
    Layout = null;
}
<!DOCTYPE html>
<html>
<head>
    <meta name="viewport" content="width=device-width" />
    <title>Index</title>
</head>
<body class="container">
    @using (Html.BeginForm("Index", "Home", FormMethod.Post))
    {
        @Html.AntiForgeryToken()
        <div id="loginbox">
            <form id="loginform" action="#" class="form-vertical">
                <div class="control-group normal_text"> <h3>Admin Login</h3></div>
                @Html.ValidationSummary(true, "", new { style = "color:red;" })
                <div class="control-group">
                    <div class="controls">
                        <div class="main_input_box">
                            <span class="add-on bg_lg"><i class="icon-user"> </i></span>
                            @Html.EditorFor(model => model.username, new { htmlAttributes = new { @class = "form-control", placeholder = "Username", @Value = ViewBag.username } })
                            @Html.ValidationMessageFor(model => model.username, "", new { style = "color:red;" })
                        </div>
                    </div>
                </div>
                <div class="control-group">
                    <div class="controls">
                        <div class="main_input_box">
                            <span class="add-on bg_ly"><i class="icon-lock"></i></span>
                            @Html.EditorFor(model => model.password, new { htmlAttributes = new { @class = "form-control", placeholder = "Password", @Value = ViewBag.password } })
                            @Html.ValidationMessageFor(model => model.password, "", new { style = "color:red;" })
                        </div>
                    </div>
                </div>
                <div class="control-group">
                    <div class="controls">
                        <div data-toggle="buttons-checkbox" class="btn-group">
                            @Html.CheckBoxFor(model => model.RememberMe)
                            @Html.LabelFor(model => model.RememberMe)
                        </div>
                    </div>
                </div>
                <div class="form-actions">
                    <span class="pull-left">
                        @Html.ActionLink("Lost password ?", "Recover", "Login", new { area = "" }, new { @class = "flip-link btn btn-info" })
                    </span>
                    <span class="pull-right"><input type="submit" value="Login" class="btn btn-success" /></span>
                </div>
            </form>
        </div>
    }
</body>
</html>