I'm aware I can't work in the US on a F2 visa. However, would it be possible to be employed by a Canadian company to work remotely from the US?
There is a discussion of the same for the person working in India the answer is overwhelmingly no. The reason is that any income from any source has to be reported on the Tax Returns that you will have to file with the IRS (yes even with the Foreign Student status), so the money you're making from your job in Canada will come up (unless of course you're going to commit tax fraud, which is a different discussion).
So having that money will violate your F-2 status.
There may be other options available such as the case similar to:
Which may allow you to own the company in the US but not receive any funds from it but whether or not this is legal, how to set such entity up, and what it would entail to your employment in Canada is a matter for lawyers, accountants, and your current employer.
Additionally, depending on where you live in Canada and your wife's school location it may be possible to get an F3 Student Visa which will allow you to live in Canada but will make your wife commute to school across the border.
There are many debates about that, and several law firms who says that this is a possibility.