Testing any non-citizen entering the US is way too intrusive. But anyone apprehended for entering the country illegally should be fingerprinted, facial and retina recognition obtained, and dna tested so the Government has all the information that could be used for confirming identity and if they commit further criminal acts it can be recognized who they are.
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You do realize that a HUGE chunk of the illegal aliens in this country actually came in legally (e.g., tourist, student, temp work) but simply never left?
Get FULL biological identity info on ALL incoming foreigners.
Overstaying one’s visa should be a felony too.