Can a Canadian citizen register a business in the US if he is under Visa work status?

Yes. A Canadian can register a business in the U.S. with OR without a visa work status. The U.S. freely allows us to register/incorporate and pay all the fees associated with creation (which is quite costly, but even more so for a non-American without an Social Security Number).

The thing that they don't mention is that once you get that business setup, invest all your time and money preparing and purchasing stock/inventory/or whatever else you'll need, if you are dealing with the public and want to actually "collect" payments for your business, it will be virtually impossible to get a merchant account that will allow you to process payments if you don't have an SSN number (which is apparently the only way they can identify your identity.) Not even online processers such as Square (who don't even require you to have a merchant account) will allow you to signup without an SSN.

Think long and hard about that before proceeding.