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>
';
}