Laid off...subcontractor advice

So I was laid off this past February due to status. A little background, I've been working there for almost 6 years, but another company bought this company last August. At the initial management meetings, they assured everything will be the same, (yea, right!)
I worked at a grocery store that was acquired by "EL SUPER" aka Bodega Latina. They are based in the LA area, but expanded over to the Inland Empire region here in SoCal.

Although I am still ticked about it, I was blessed to have an income for so many years and allowed me to pay for my bachelor's degree. I was reading other threads and came to an idea of becoming a subcontractor, but I have a few questions.

Should you just ask a business to hire you as a subcontractor rather than employee?
Business name, any recommendations? I was thinking being my name "John Doe Co"

Lastly, just want more feedback or experiences with subcontracting.

I personally know two people who went through this path. One is working for a $100 billion/year multinational corporation as a contractor(But really an employee in practice) making $65K/year.

So while this can be done, the employer must be willing to do this for you, and you have to be compelling enough for the employer to retain you this way. Having connections inside the employer helps a lot.

I am about to do this my self. I am an engineer student and I am currently working on the union payroll as anyone else. They notified me about the ss# problem last week. I was brainstorming with human resources and they decided that subcontractor would be the way to go. I must admit that getting to know people and getting involved has helped me a lot. I am super flexible and I was able to learn to work in different departments, sales, engineering, IT, I have even done receptionist and remodeling layouts. Now I am officially at engineering. I am still in my senior year so we will see how it goes. The best way to go about this is to know someone within a company. Family, friends etc.

Thanks for the input guys. Although I am now working, I am looking into starting my own business. Well see how it goes!