First of all there is no Canadian GC. If you become a Canadian PR, you get a "Maple Card". As far as I know, an individual is allowed to apply for asylum in either the US or Canada. Both countires, I believe, have an agreement to stop individuals from applying for asylum in both. However, if an individual fails to gain asylum in the US, he/she is probably allowed to seek non-asylum based PR in Canada. I may be totally wrong however. Consult a lawyer on this.