Other one was for knowledge purposes.

Then I would suggest mine as there's no point in assigning a value which is already boolean to string using tertiary if. That just produce messy code. You can convert using .toString()

eg:-

chkbox.IsChecked.toString();

    command.Parameters.AddWithValue("@ParamNameHere", valuehere);
    command.Parameters.AddWithValue("@CheckBoxChecked", chkbox.IsChecked);

  @ParamNameHere and @CheckBoxChecked ?

using (SqlConnection connection = new SqlConnection("ConnectionStringHere"))
{
    using (SqlCommand command = new SqlCommand())
    {
        command.Connection = connection;            // <== lacking
        command.CommandType = CommandType.Text;
        command.CommandText = "SQL Statment here (insert or update)";
        command.Parameters.AddWithValue("@ParamNameHere", valuehere);
        command.Parameters.AddWithValue("@CheckBoxChecked", chkbox.IsChecked);

        try
        {
            connection.Open();
            command.ExecuteNonQuery();
        }
        catch(SqlException)
        {
            //error here
        }
        finally
        {
            connection.Close();
        }
    }
}

above is the correct code to save the data to db

if (checkBox.IsChecked)
{
  MessageBox.Show("CheckBox ticked");
}
else 
{
  MessageBox.Show("CheckBox unticked");
}

So here if the checkbox has been checked then it will display Checkbox ticked else checkbox unticked. You don't have to go round and round using tertiary ifs and another variable to get its bool value.