If you are applying outland, you don't have legal status in Canada. You are a visitor each and every time you cross the border, up until you have PR status. Therefore, you shouldn't need proof of legal status in Canada for your application.
I'm assuming you're from the US based on your post. Why do you need them to stamp your passport? What is the benefit? If you are crossing a land border between the USA and Canada, you do not get a stamp unless you are not granted the full 6 month visit.
To your original question - do not over stay in Canada. While you may not have heard of it causing issues, there is always a chance it can cause an issue.