Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions. First of all, know that living in a non-Muslim country is not permissible for it has many ill effects on a Muslim's religion and on his whole life. The act of taking the nationality of a non-Muslim country is similar to the act of living there. It is even more offensive from living there only. For more details, please refer to Fatwa:82552.
Therefore, if you are not among those persons who are permitted to take the nationality of a non-Muslim country for legal reasons, then you are neither permitted to take the USA nationality nor to lie for that purpose. If you are compelled to take this nationality and your condition legally allows you to receive this nationality, as mentioned in the above Fatwa, then you may take it. But lying is not permitted in Shariah and a Muslim should not ascribe to it. So, if there is no chance to get what you desperately need except by lying, then you may use allusive expressions to avoid outright lying.
Allah knows best.