There are no performance implications since the compiler will translate your lambda expression into an equivalent delegate. Lambda expressions are nothing more than a language feature that the compiler translates into the exact same code that you are used to working with.
The compiler will convert the code you have to something like this:
public partial class MyPage : System.Web.UI.Page
{
protected void Page_Load(object sender, EventArgs e)
{
//snip
MyButton.Click += new EventHandler(delegate (Object o, EventArgs a)
{
//snip
});
}
}