False. They only took people before they were drafted. Once the person here received a actual draft order by the military, Canada did not help them (not officially, anyway).
Not true. Canada turned a blind eye to draft dodgers because draft evasion is not a crime here. Official records show that as many as 30,000 draft evaders came up here, but unofficially many more slipped across the border without declaring themselves because Customs and Immigration officials were instructed not to question people about their draft status. After the amnesty was declared in 1977, about half of them returned to the U.S., but the other half stayed here and most became Canadian citizens.
Deserters were another matter because desertion is a crime here. Our military condemned the practice, but ended up looking the other way, despite pressure from the U.S. military. Subsequently, our government only paid minimal attention to it (i.e. nobody was charged with an offence and/or deported). The only times that deserters were charged were when they voluntarily returned to the U.S.
On a personal note, I know many draft evaders and a few deserters because my family is originally from a town in southeastern B.C. where we welcomed them with open arms. We already had an interesting mix of First Nations, hippies, Doukhobors, descendants of WW2 Japanese-Canadian internees, Mennonites, loggers, and farmers, and were known as a haven for pacifists, so an influx of hundreds of people resisting a war was considered fairly normal. As a result, we all got along and it wasn't surprising when many of these men (and their families) also became Canadian citizens.