Using this function you are able to detect the presence of ?video=1 in the url:
function getURLParameter(name) {
    return decodeURI(
        (RegExp(name + '=' + '(.+?)(&|$)').exec(location.search)||[,null])[1]
    );
}
Source: Get escaped URL parameter
Credit to https://stackoverflow.com/users/726427/pauloppenheim
Then you could do something like:
if(getURLParameter('video')==1){
  $(".showVideo").trigger('click');
}
edit:
$(document).ready(function(){               
    function getURLParameter(name) {
        return decodeURI(
            (RegExp(name + '=' + '(.+?)(&|$)').exec(location.search)||[,null])[1]
        );
    }
    if(getURLParameter('video')==1){
      $(".showVideo").trigger('click');
    }
});
Wrap the parameter name(video) in quotes getURLParameter('video').
Another Edit
Wrap your click event handler in a function, basically remove everything form:  
$('.showVideo').live('click', function() {
    //build overlay
    (...)
    return false;
});
Cut&paste it inside a function. Then just call the function from inside:
$('.showVideo').live('click', function() {
    my_function();
});
Then change the previous code to:
if(getURLParameter('video')==1){
     my_function()
}