Yes, as noted above, there are certainly some jobs with the federal government (and I believe some at provincial and other levels) that can require (by law) citizenship or PR status. And obviously for those that specifically require it, they can ask for proof. (How they check a card or passport is not forged, no idea - common sense I should think, or submit info to other parts of government). Note of course that some of these jobs may involve security clearances and background checks different from normal private sector or even government jobs.
BUT: I would also not assume that just because 'government' asks to see a PR card or passport, that they automatically have a right to do so as an employer. Government is made up of people who often don't know all applicable rules any better than anyone else, and worse, can sometimes assume that some rules do not apply to government - because government. That said, I don't know in this particular case (about what ID they can ask for).
At any rate, OP was referring to small businesses, so not the case for this inquiry.