Suppose you have the breeders per string, something like
$div[0]='<div>My breeder 1</div>';
$div[1]='<div>My breeder 2</div>';
$div[2]='<div>My breeder 3</div>';
$div[3]='<div>My breeder 4</div>';
and an array
$my_breeders=[0,1,2,3];
If you shuffle the array, you will get something like
$my_breeders=[1,3,2,0];
this is the order you display your divs in. Put this in a string and write it to a cookie.
$order=implode('-',$my_breeders); //gives a string: 1-3-2-0
setcookie('breederorder',$order,time()+(30*60) )
the next time someone visits the page, check for the cookie
if(!empty($_COOKIE['breederorder'])){
$my_breeders=explode('-',$_COOKIE['breederorder']);
}
Now you have the same array as before.
Note, I set the cookie time at half an hour (30*60). If you set it too long, the next time a visitor comes to your page, het will still get the same order.