Is it halachically permissible to hire an illegal alien, for example, as a maid?

It’s definitely not allowed. There are so many Jewish brothers and sisters, many of them new olim, in need of work, and one should clearly give them preference. In addition, one must follow the law of the land, dina d’malchuta dina, and not support those who are here illegally. With Love of Israel, Rav Ari Shvat