Here is something to consider.
Illegal immigrants are not citizens of the united states. An argument can (and has) been made that they are still protected by the Federal Constitution. Since the courts have ruled that illegal immigrant prosecution and deportation is exclusively at the privilege of the fed, then Arizona could detain them as criminals (no legal argument their) but not have the power to prosecute them or allow them habeus corpes.
Logically, they can be detained as criminals but not charged as they are not citizens and the state has no jurisdiction to prosecute but a duty to apprehend them.
You can't have it both ways. They can't have constitutional rights, and not be allowed habeus corpus, which is a constitutional right.
Btw, just to beat anyone to it, aliens do indeed have constitutional rights.
If aliens had no rights under the Constitution, they might not only be banished, but even capitally punished, without a jury or the other incidents to a fair trial. But so far has a contrary principle been carried, in every part of the United States, that except on charges of treason, an alien has, besides all the common privileges, the special one of being tried by a jury, of which one-half may be also aliens.--James Madison