It is "most" likely Ich.
They are small parasites which will attach to the fish's weakened slime coat, and then jump off to breed and multiply, and then attach back to the host again.
That is likely why you see them going from fish to fish.
Best thing to do is raise the temperature to 81F and add some aquarium salts (around 1tbs per 5 gallons) to thicken their slime coat and resistance.
If they attack your fish again, go buy an Ich treatment, like Quick Cure, or such, and the raised temperture will force their breeding cycle to proceed more quickly, and when they detach from the fish, they'll be killed while free-swimming.
They can kill your fish if allowed to overpopulate, so make sure to handle it quickly.
There's a few other things the "white spots" can be, but 90% of the time in recently established tanks, it is just a case of Ich.
Think of them as fleas on a dog--except they're on your fish.