I've always used high temp hose. Your cooling water should protect it during normal use regardless but by going high temp you have a bit of added insurance. You can also circulate boiling water to sanitize without the cooling water running and not have any worries.
When building a counterflow chiller should the garden hose be of the high temp kind?
No. Ordinary garden hose will work just fine. I would get a real rubber hose and avoid the cheaper vinyl stuff. The rubber is more durable and the last thing you want is to have to replace the hose for whatever reason. The highest temp it could possibly see would be near 212F and that would only be when it is not under pressure with no cooling water flowing. The 212F won't melt the hose and it won't catch on fire. Absolutely nothing to be concerned about.