True, the Bill of Rights applies to everyone, even illegal immigrants. So an immigrant, legal or illegal, prosecuted under the criminal code has the right to
due process, a speedy and public trial, and other rights protected by the
Fifth and
Sixth Amendments.
This fact sheet from the National Lawyers Guild outlines a host of rights afforded to immigrants and citizens alike. (There are a few rights reserved for citizens. Among them are the right to vote, the right to hold most federal jobs, and the right to run for political office.)