They absolutely do. There are cheats and hacks you can use to make decent-looking poached eggs regardless (molds, vinegar and salt in the poaching water, double

They absolutely do. There are cheats and hacks you can use to make decent-looking poached eggs regardless (molds, vinegar and salt in the poaching water, double-cooking with an intermediate ice bath, straining off the weaker outer proteins before cooking, ...), but fresh eggs give a more consistently good result.