How can I get this code to only output one link?

You need to break both foreach loops not just one, so add break 2; after the echo.

   $mycats = array(106, 107);
    foreach (get_the_category() as $childcat) {
        foreach ($mycats as $mycat) {
            if (cat_is_ancestor_of($mycat, $childcat)) {
                echo '<a href="'.get_category_link($childcat->cat_ID).'">' . $childcat->cat_name . '</a>';
                break 2;
            }
        }
    }