Canadian quarters will not operate the machine, but Canadian pennies will press in the machines.
You do want to use as clean a penny as possible because you get a better press. Also, some people prefer to use only pennies that a fully copper. US pennies started using copper-clad zinc as pennies in 1982; Canada started in in 1997. If you use one of these more recent pennies, you can get "silver" streaks in the penny as the zinc shows through on the press. Some people aren't bothered by that. I am. I only use pre-1982 US pennies.
As for whether the penny you put in is the one you get out, I can say with confidence after collecting a couple of hundred, that you get what you put in. You put in a dirty penny, you will get a dirty imprint. Put in a 1984 penny, you will get silver streaks. Put in a 1981 penny, you will not. You can even see the faint imprint of the raised part of the penny on your pressed version. Put in a Canadian penny, you will see the Canadian imprint. You do not get a fake penny.
Copper is a soft metal that is easily pressed.