I would suggest that the OP look at the career websites for the companies that have MECH E’s and see if they would be hiring/sponsoring non-citizens. My daughter’s company has it posted on their website that they can’t sponsor non-US citizens.
That’s different than needing security clearance, which was the issue I was speaking to. Outside of aerospace, and specifically those jobs dealing with defense, I’d guess security clearance need to be rare.
I agree the need for a security clearance is fairly rare in engineering and is only common in certain fields. It is also quite common for companies to hire foreign students/engineers provided the work is not export controlled.