You cannot work in the US (i.e. the work is performed there) on an H-4 visa. Or without specific work authorization.
The income types you are talking about are not the same as employment income (well income from stocks isn't). Besides that's a tax issue, not an immigration issue, the two are separate.
Rental income from Canada is Canadian-source income effectively connected with a trade or business (as the new tax treaty stands) but the employment (landlord duties) in order to receive the income is in Canada. No physical work is performed in the US so there is no need for work authorization.
It's where you physically are when the work is performed that matters for immigration issues (and many tax issues because he still has to declare the Canadian-source income in the US).
For example if he worked in telesales, it's where he is when he answers the phone that matters.
I don't think it's ever actually come up in US courts but it's come up in
UK courts when people have been on holiday there and the boss has phoned them to ask a question - technically they are then working in the UK and need work authorization. The court applies the principal of
de minimus non curat lex, i.e. the "law is not concerned with trifles". As a result caselaw is that for immigration and tax purposes (according to the interpretation by the authorities) in the UK you can be there for up to two weeks doing something connected with work (training, meetings) without needing work authorization or filing tax paperwork.
But none of that helps you of course, sorry. Might want to try getting H-2B for him is the only advice I can offer.
Steve.