Although people have already replied to you stating (accurately) that you can be on the board of a company no problem, keep in mind that "being on the board" and "getting paid" are going to be vastly different things; the moment money is brought into the question there are a lot more problems.
For example, you say you're coming to the US soon, what does that mean? I assume, since you mentioned Stanford, you'll be coming in as an F1 (student) visa. The work restrictions placed on an F1 visa are very strict, in order to be gainfully employed you'll have to go either through CPT work allowance (which you can do while you're a student), or OPT work allowance (which can happen before and after you graduate). Your company must be registered in the E-Verify program and be able to prove to the US government that your employment is directly related to your field of study. You'll have to get paperwork emitted and signed by your school's DSO, your company, and potentially a faculty member or advisor.
Furthermore, all employment authorizations tied to an F1 visa are time-limited, you'll have to convert to a different visa which allows you to live and work in the US after you graduate. If you're an executive of the company, they may be able to bring you over under an L visa which you can convert to a green card down the line. You can also go the H1B route, which is longer, more difficult, and has a much higher likelihood of failure (as it's a lottery).
All that being said, my main point is: you can be named any position you want in the company, but if you want physically live in the US and get paid to be in this position, you'll have to go through an extensive immigration process. I highly recommend hiring an immigration attorney.