mcp111 says:

Then why textbox.text returns null?

 The value is assigned at client side to TextBox. So you are getting null.

To get the value Request.Form collection is used.

Refer below sample.

HTML

<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script src="Scripts/jquery.dynDateTime.min.js" type="text/javascript"></script>
<script src="Scripts/calendar-en.min.js" type="text/javascript"></script>
<link href="Styles/calendar-blue.css" rel="stylesheet" type="text/css" />
<link href="Styles/calendar-blue.css" rel="stylesheet" type="text/css" />
<script type="text/javascript">
    $(document).ready(function () {
        $(".Calender").dynDateTime({
            showsTime: true,
            ifFormat: "%Y/%m/%d %H:%M",
            daFormat: "%l;%M %p, %e %m,  %Y",
            align: "BR",
            electric: false,
            singleClick: false,
            displayArea: ".siblings('.dtcDisplayArea')",
            button: ".next()"
        });
    });
</script>
<div>
    <asp:TextBox ID="txtDOB" runat="server" ReadOnly="true" class="Calender" />
    <img src="calender.png" />
    <br />
    <asp:Button ID="btnSave" runat="server" Text="Save" OnClick="btnSave_Click" />
</div>

Code

C#

protected void btnSave_Click(object sender, EventArgs e)
{
    DateTime dob = Convert.ToDateTime(Request.Form["txtDOB"]);
    ClientScript.RegisterStartupScript(this.GetType(), "alert", "alert('" + dob + "')", true);
}

VB.Net

Protected Sub btnSave_Click(ByVal sender As Object, ByVal e As EventArgs)
    Dim dob As DateTime = Convert.ToDateTime(Request.Form("txtDOB"))
    ClientScript.RegisterStartupScript(Me.GetType(), "alert", "alert('" & dob & "')", True)
End Sub

Screenshot