[So, are we worried about the Constitutional rights of illegal aliens here, or what?]
No, we are worried about the rule of law.  Even more so when it involves someone who was invested with the power of authority and the use of violence to enforce that authority.
A civil society functions on either a general acceptance of abiding by the rules or else by threat of the fall of the sword.  In other words we either follow the rule of law or fall into lawlessness.