Forming a US company with a Canadian citizen
It would be no problem for you to register a U.S. company together with a Canadian Citizen. The structure of the company would depend largely on the type of business you would be conducting. Typically, LLC's are often preferred as they are flexible and easier to maintain. We have a research article titled 'LLC vs. Corporation' which offers a side-by-side comparison of the entity types.
Your partner would not need to supply a social security number to form the business or obtain the federal tax id for the business (we can use your number for that matter). If your partner is going to be reporting U.S. income to theIRS, it is likely he will need an ITIN (individual taxpayer identification number). This is always best to consult with a licensed accountant.
We offer a free tax-webinar that addresses many of the common taxation questions of operating a U.S. business, as well as offers resources to accounting and other financial assistance.