WordPress plugin: admin-ajax.php not passing data to custom function

As czerspalace mentioned you need to pass the action to the .ajax().

One way to do this would be to modify your form:

function ea_display_form()
{
    echo '
        <form id="eaForm">
            <input type="text" name="username" />
            <input type="submit" value="Submit" />
            <input type="hidden" name="action" value="ea_ajax_handler" />
        </form>
        ';
}