No, and we recently spoke about a case involving exactly this issue. While an employer can confirm that you are legally eligible to work in Canada, they cannot discriminate against you based on your citizenship.
In this example, a student in Canada on a temporary visa had lied on their application form and said they were permanently eligible to work in Canada because they knew that they would be denied an offer if they were truthful. The Ontario Human Rights Tribunal ultimately awarded the potential employee significant damages, both to cover wages that he would have earned if he had been rightfully hired in the position, as well as damages for the injury to his dignity (which some human rights tribunals are empowered to compensate).
Ultimately citizenship is a protected ground like any other, and an employer does not need to know yours to confirm that you are legally eligible to work in Canada. We advise all of our employer clients as a matter of course to avoid asking for unnecessary personal information which can later be used as a basis for discrimination.
