Perhaps I'm reading the question wrong, but it appears that it's directed at MY conduct, not the ones who have entered illegally.
Clearly, those who enter this country illegally are not following God's will. But what is my role as enforcer?
I don't really have an answer, other than treat them kindly and respectfully. We should help them to become compliant with our laws, and if that means that they have to leave for a time before they can be legally here, well then that makes sense to me.
However, it really surprises me how so many people look upon illegal aliens as sub-humans. Many of them are here in order to better provide for their family. I can respect their goals, but not ther methods.
One perspective if the problem is to recognize God has provided four divinely established institutions for believer and unbeliever alike. Whatever God creates and establishes was performed with perfect judgment and righteousness.
Those four institutions we are revealed in Scripture to be Volition, marriage, family, and national governance.
It should also be recognized that living in violation of these institutions or attacking their basis is merely a losing operation.
With that said, for those nations which followed the will and plan of God, there also exists privileges associated with a client nation status. Whenever a client nation steps away from God, though, cycles of divine discipline are in store for the disobedient nation. If the first cycle of discipline doesn't result in the client nation returning to God, another cycle, seven times more severe is meted out as a second cycle of discipline.
In 2nd and 3rd cycles of discipline, a nation may be subjected to the will of its enemies, but still with opportunity to return to Him and regain favor.
Now with respect to the illegal immigrant, let's be a bit more specific here and refer to illegal immigration from Mexico by an impoverished people seeking more worldly success in the US.
We know from Scripture there are angels who are assigned territories and possibly nations. During the Church Age, humankind works according to the plan of God in assisting with resolving the angelic conflict in the battle of good and evil. Sin has already been resolved as an eternal issue at the Cross. Good and evil, though must be resolved in time. We are instructed to obey legitimate authority.
Part of this obedience is to insure a proper testing of any and all counterfeit advances upon God's plan by the Adversary. Although the results are foreknown, the testing continues.
For the believer and unbeliever, remaining within those four institutions is a safe harbor for man, while that testing continues.
Not respecting the sovereignty of national borders means one has violated the relationship of land with governance as an attribute of the nation. If limited between 2 nations, it means they are joining one another.
As an aside, from the perspective of theological 'Date and Codd' in RDBS Algebra/Calculus we might express this differently. Declaratively, if we look at any and all issues that might be in the middle of testing amongst the angelic host in one nation and now join them to those of a second nation, we've performed a Cartesian Product of those relationships. This means we've opened up a bag of worms that neither nation was probably ready to even begin addressing or identifying, let alone solve righteously, simply by the vast number of relationships being posed.
Next, let's open the door a bit further and allow more nations into the fray. The exponential raising of issues can easily lead to lack of respect for any authority, let alone any single government.
This situation to promote illegal immigration then tends to follow a path of rebellion towards legitimate authority rather than respectful subjection to proper authority. Accordingly, during this Church Age isn't a legitimate course.
Now for a counterargument. No matter what the environment, God has provided a perfect problem solving device for man by the means of faith in Christ. No matter how heinous the situation, if a believer simply obeys God through faith in Him, he will be provided for by God. Not necessarily in a worldly or fleshly fashion, but God has a plan for all of us. There may be many situations where a poor faithful servant of God ventures north of our southern border in accordance with His plan.
Discernment then becomes critical. Not to become too legalistic and worldly, but also not abdicating one of His established institutions for man. As a nation, we might experience a cycle of discipline where His plan results in our losing some land to even our enemies. All the more reason to turn away from disobedience, return to Him and place faith in Him.