Would you have police approaching people on the street, based on a suspicion that they may not have legal status?
Absolutely. In times of war in the past, soldiers wore uniforms for a reason. So the enemy could be identified. And if they wore the same uniform their accent or lack of language understanding was a dead give away that they were enemies.
These illegals broke a fundamental law. Who cares about profiling. The majority of illegals happen to be Mexican. So they will speak broken English or won't even be able to communicate in English and will obviously look like a Mexican with dark hair, darker skin etc. Most officers will have a good idea if they are looking at an illegal immigrant or not. And if they happen to have legal status, good for them.
I love how everyone keeps scratching their head saying "How are you going to get them out?" We're not trying to send them to Mars!! This is not fucking rocket science. As Pray said, walking is cheap.