How to write a regular expression which will accept all character except ' and "

... new solution ...

To both prevent the user entering those characters at run-time, and catch the 'Paste event and remove those characters, you can use a UserControl:

using System.Text;
using System.Windows.Forms;

namespace YourNameSpace
{
    public partial class NoQuoteTextBox : TextBox
    {
        private const int WM_PASTE = 0x0302;

        private StringBuilder sb;

        public NoQuoteTextBox()
        {
            InitializeComponent();
        }

        protected override void WndProc(ref Message msg)
        {
            if (msg.Msg != WM_PASTE)
            {
                // good to go here
                base.WndProc(ref msg);
            }
            else
            {
                // handle paste
                sb = new StringBuilder(Clipboard.GetText());

                if (sb.Length > 0)
                {
                    sb.Replace("\'", string.Empty);
                    sb.Replace("\"", string.Empty);

                    if (sb.Length > 0)
                    {
                        Clipboard.SetText(sb.ToString());
                        base.WndProc(ref msg);
                    }
                }
            }
        }

        // handle user keyboard entry at run-time
        private void NoQuoteTextBox_KeyPress(object sender, KeyPressEventArgs e)
        {
            e.Handled = e.KeyChar == '\'' || e.KeyChar == '\"';
        }
    }
}

... previous solution ...

Why not prevent those characters from being entered ?

private void SomeTextBox_KeyPress(object sender, KeyPressEventArgs e)
{
    if (e.KeyChar == '\'' || e.KeyChar == '\"')
    {
        e.Handled = true;
    }
}