Remember General Eisenhower? You know the General that saved your sorry butts form the Nazis? Well he became President and engineered Operation Wetback in 1954. Look it up Mr. Stiff Upper Lip.
I looked up operation Wetback. It was an unmitigated PR disaster, and it ran into the problem that deported mexicans would just cross back over into the US again. What few successes it had were because the Mexican government (who largely initiated the program) were very supportative of it. That is not the case anymore, and it unlikely to be so for quite a few years.