The body of evidence requiring deportation is that you're not here legally. The absence of your legal status is all the habeas corpus necessary.
While I am always inclined to doubt Trump and his advisor's good faith, it is possible that what they are looking at is more nuanced that a blanket suspension. Migrants can have one of several status's and there may be some that are exempt and others the administration believes should not be exempt.
For example, a migrant may or may not have entered illegally. And they, even if illegal, may or may not have subsequently been given permission to temporarily stay (eg Biden's parole). In addition, many may have gotten temporary permission to stay but dodged their hearings (500,000 in last year). And illegal migrants caught within 100 miles of the border are subject to deportation immediately, but not those who escape deeper into the US.
It would be logical (but not necessarily lawful) to deny habeas status to those who entered illegally, worked illegally, and were caught. But it's not so much to those who are long time permanent residents with a clean record who have broken no law. Any removal of habeas in the latter category would surely fail in court.
Still the courts have found that, depending on circumstances, deportation is not a criminal punishment but an administrative process and solely within the purview of Congress and the President. Hence, there is room to argue that habeas may not apply as broadly as assumed.
However, as usual with Trump and his people, they love to rush in, operate outside the law, in highly irregular ways sure to create a judicial confrontation. In other words, as is typical of Trump, he'd rather pick a fight over a micro percent of the migrant population, one he is destined to lose, while ignoring what is actually needed. In short the removal of Kilmar Abrego Garcia to El Salvador was illegal, as was his use of the Alien Enemy Act was illegal. He accomplished nothing.
If you want to address how we got up to 20 million unlawful migrants in the US it is due to the failure of courts to enforce the laws we have, and a failure to demand from Congress the money necessary to enforce those laws AS written. Yes, some laws need streamlining or clarification as well, but none of this is possible when the administration wastes everyone's time making martyrs instead of proposing reform. Address those migrants who are totally illegal, address those who have committed felonies, address those that fail to appear at hearings. Identify potentially winnable court cases that might significantly change case law and improve a Presidents option.
But calling the judges communists and threatening Constitutional crisis is a losing proposition.
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