My 4 y.o. border collie was perfectly fine this a.m. and now he can't stop urinating, though nothing is coming
out at this point, he has the constant urge and is asking to go out every 10 minutes. I noticed he hasn't been drinking his water lately, but then he likes to eat the snow. I'm worried. Could his kidneys be effected by him playing outside in the snow?
I'd be more inclined to think "UTI" or "bladder infection" than "kidney disease". If he's decreased his water intake because he'd rather eat the snow, there may be a urine production decrease that could make it easier for bacteria to get a foothold in his urinary tract.
Contact your vet and see when you can get an appointment - the sooner the better. And encourage your dog to drink.