Just to be clear on your last point, I don't object to immigrants pursuing engineering, at all, provided it was a choice they made freely, that the immigrants were also free to choose not to become engineers as a condition of entering the US. I do object to empowering employers to coerce would-be immigrants into studying engineering (or any field for that matter) or working as engineers as a condition of gaining access to the US labor market, as I believe that this produces harmful market distortions and is an affront to personal freedom.
Generally speaking, immigrants (about 1.2 million annually to the US) are free and full participants in the labor market, free to pursue the degrees and skills they wish, in response to salary, working conditions, career longevity, personal interests, and so forth. They may wish to open a sandwich shop, sell real estate, or write software. Just like people who were born in the US.
It turns out that, like those born in the US, these immigrants don't go into engineering in numbers that high tech employers feel they should. Personally, I think this is a sign that perhaps high tech employers need to sweeten the pot a bit - if people (immigrants or otherwise) who are free to choose aren't choosing you, that's the market's answer. It's not them, it's you.
Employers have responded by lobbying for what I consider to be a wildly self-serving and coercive visa that gives them the power to bestow and revoke US residency and work rights under the notion that there is a "shortage" of engineers. This enables them to say - we'll let you in if you study computer science and agree to write code for us for the "market rate" salary that fails to attract those with the freedom to choose their careers in sufficient numbers. We won't let you in if you don't take our tech test, and if you try to quit your job and open up a sandwich shop, we'll have you deported. You know, free labor markets.
I'm opposed to a system that allows employers to bestow (and revoke!) US residency and work rights on non-citizens under the condition that they study what the employer says they should study, work on what the employer says they should work on, live where the employer says they should live, and so forth. I think this position is very consistent with pro-immigration attitudes and personal freedom.