I guess the bottomline is that you can start and invest in a company, but not work for it.
Have you already applied for a Green Card? While you wait, maybe you could invest your free time now in an open source project that you could potentially leverage for-profit services later on, after you get your GC.
Another thing you could do while you wait for your GC is to simply start a savings account and accumulate a little bit every month, so when you get your GC you have enough funds to start a company and survive without a salary for a while (I'd recommend 1 year of living expenses).
Another option is to go get some higher education, which you can do as a H1 / work visa. Because you work for a large company, they probably offer tuition reimbursement programs. Take advantage of those programs as much as you can!
In short, use the time between now and your GC to prepare yourself and your life to eventually start a company. I'm now a GC holder, but I have immigrated to US via a work visa as well... So I understand your frustration, but I don't think you have many options, unfortunately.