I live in the united stated. I will be called to serve jury duty. If the judge instructs to ignore some evidence and follow others regarding a particular case, for example, the police entered the criminal's house without a permit from the judge but they found the object used to commit the crime. Therefore, the evidence is rejected. What will be my approach in making a decision following the judge's instructions and doing the right thing at the same time?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Whenever you give a verdict against a person or in favor of a person, it must be according to Islamic Shari'a. Giving rulings not in accordance with Shari'a is not permissible even if the convicts are non-Muslims.
Allah Says (interpretation of meaning): {… So if they come to you (O Muhammad (Sallallahu Alaihi wa Sallam)), either judge between them, or turn away from them. If you turn away from them, they cannot hurt you in the least. And if you judge, judge with justice between them. Verily, Allah loves those who act justly.} [5: 42].
Imam al-Qurtubi said: 'Whenever one judges among non-Muslims should implement the rulings of Shari'a'.
Imam Ibn Taymiyah said: 'No one is permitted to give rulings to Muslims or non-Muslims other than the rulings of Shari'a. If one gives rulings other than Islamic Shari'a he comes under the Saying of Allah: {Do they then seek the judgment of (the days of) ignorance? And who is better in judgment than Allah for a people who have firm Faith.} [5: 50].
It is known fact that all courts in non-Muslim countries act according to the civil law that is derived from false religions. Thus, they are contrary to Islamic Shari'a.
Therefore, in the mentioned job you will be subjected to those civilian laws and somebody else will join with you in taking decision. So, it would be impossible for you to pass a judgment that conforms to Shari'a.
In addition, you have to give your rulings in matters that are forbidden in Shari'a such as alcohol, pork, usury and the usury-based organizations, etc. Thus it is forbidden for you to accept the above job.
Allah knows best.