IANAL but based on experience, in general you can legally "own" a company/startup while on F1/H1B but you cannot "run" it. However, USCIS has come up with an explanation of "Employer-Employee relationship [0]" which is worth looking into if on H1B. The Requirement 1 is interesting:

    + How do I demonstrate an employer-employee relationship   if I own my own company? 
    If you own your company you may be able to demonstrate  that an employer-employee relationship exists if the control  of your work is exercised by others.  For example, if your company has a board of directors, preferred shareholders, investors, or other factors that show your organization has the right to control the terms and conditions of your employment (namely the right to hire, fire, pay, supervise or otherwise control the terms and conditions of your employment), then you may be able to meet this requirement.   Some of the evidence you may submit to demonstrate the distinction between your ownership interest and the right to  control your employment includes:                

    Term Sheet
    Capitalization Table
    Stock purchase Agreement
    Investor rights Agreement
    Voting Agreement
    Organizational documents and operating agreements

Otherwise, your options are almost none unless you quality for EB5 [1]

[0] http://www.uscis.gov/eir/visa-guide/h-1b-specialty-occupatio...

[1] http://www.uscis.gov/working-united-states/permanent-workers...