There are a few steps to this, as I found out today. First, you add a custom image size (you can put this somewhere the functions.php file in your theme):

add_theme_support( \'post-thumbnails\' );
add_image_size( \'my-size\', 550, 550 ); // not sure what you want your height to
                                     // be, but this\'ll shrink it to 550px
                                     // wide *or* tall. If you want it to be
                                     // cropped to a 550px square, add true as
                                     // a fourth argument

The unfortunate thing is that WordPress doesn\'t actually show this new size in the media uploader. You\'ll have to put a function that does this into your functions.php as well (source: http://kucrut.org/insert-image-with-custom-size-into-post/):

function my_insert_custom_image_sizes( $sizes ) {
    // get the custom image sizes
    global $_wp_additional_image_sizes;
    // if there are none, just return the built-in sizes
    if ( empty( $_wp_additional_image_sizes ) )
        return $sizes;

    // add all the custom sizes to the built-in sizes
    foreach ( $_wp_additional_image_sizes as $id => $data ) {
        // take the size ID (e.g., \'my-name\'), replace hyphens with spaces,
        // and capitalise the first letter of each word
        if ( !isset($sizes[$id]) )
            $sizes[$id] = ucfirst( str_replace( \'-\', \' \', $id ) );
    }

    return $sizes;
}

// apply the above function as a filter on the media uploader\'s list of
// image sizes
add_filter( \'image_size_names_choose\', \'my_insert_custom_image_sizes\' );

Lastly, to set this size as the default, you\'ll need to change a WordPress setting. Add a filter on the setting in question. This has the benefit of being portable with your theme in which you\'ve defined all the above options.

function my_set_default_image_size () {
    return \'my-size\';
}
add_filter( \'pre_option_image_default_size\', \'my_set_default_image_size\' );

I tested this all out on WordPress 3.4.2, and it seems to work great.

Note: I\'d recommend putting the above PHP code into an init function of some sort and attaching it to the \'after_setup_theme\' hook:

// define the functions first
function my_insert_custom_image_sizes( $sizes ) {
    // not going to repeat the function body, for brevity\'s sake
}

function my_set_default_image_size() {
    // ditto
}

// define the init function next, which sets up all the necessary stuff
function custom_image_setup () {
    add_theme_support( \'post-thumbnails\' );
    add_image_size( \'my-size\', 550, 550 );
    add_filter( \'image_size_names_choose\', \'my_insert_custom_image_sizes\' );
    add_filter( \'pre_option_image_default_size\', \'my_set_default_image_size\' );
}

// and attach that init function to the \'after_setup_theme\' hook
// so it runs on each page load once your theme\'s been loaded
add_action( \'after_setup_theme\', \'custom_image_setup\' );