My friend and I want to swap cars for a day. Is this ok in the eyes of
the law and our insurance companies? (We're in the US)

I beleive that it is OK, as long as you are not planning to do
anything illegal with your vehicles. If your friend gets stopped by a
police officer in your vehicle he propably will not have any trouble.
However, depending on your insurance policy if your friend takes your
car and gets into an auto accident where he is at fault then your
insurance agency may not cover damages to the vehicle, him and/or the
victm or his property. It is not illegal to switch cars, as long as
you do not intend to do it for a long period of time, and do not
report your vehicle stolen.

Isn't it typically illegal to drive w/o insurance though?

... or would his insurance co. typically cover him?

The answer to your question is going to depend upon the terms of your
insurance contracts, which makes this question difficult to answer
with any degree of certainty.

As expertlaw said, the contract terms in the policy are what will let
you know.  For example, my insurance covers "any licensed driver" in
my car and I am also covered when I drive "any vehicle" my drivers
license grade allows me to operate.  If you have this you should be
fine.  I would suggest that if you do trade cars, you have with you
the proof of insurance for the car you will actually be driving.  So
trade the insurance cards in your wallet along with the keys.

Cool- very helpful.. thanks!!