Can One Start A Company or a home business in USA on H1 ?

Yes, any one can start a company or business on H1 or on any other Visa category in USA, In fact one can start a company or business in USA even if they do not have any Visa to USA or do not intend to go to USA at all, but want to perform business there.

There are certain pre-requisites that have to be met before you can open a company. like having a lease of office space etc. The same can be achieved by registering  with virtual office providers.

Also, kindly understand There are Tax implications and local state government registrations that have to be performed for and after opening a company in US.

It is very important that all formalities are met before you initiate your business in USA.