Yes you can.
There will be more moisture in the air, so if you're getting condensation on the car, it won't work well.
Also, you'll need some lighting, so that you can see where you've covered and where you haven't, and you need light to make sure you get all the wax off.
Keep in mind though, that lights at night attract bugs. Don't have a light directly over the car if you're doing the waxing outside. You'll attract bugs, which will then get zapped, and you'll have dead gnats and other annoying pests all over your car.
Swirls are not created by waxing at night. Most likely, you were waxing a dirty/dusty car (but you couldn't tell due to lack of light), using a cheap, coarse towel or T-shirt to apply the wax, or using the above to remove the wax.
70*F is not cold. If it were to hit 70*F here tomorrow, I would be outside detailing my car from sunrise to sunset!