Well, the problem with waiting is that people (kitty or human) with chronic runs will get dehydrated fast. If it persists for even as short a time as 24 hours, their bodies will not have had the time to absorb the water they've been drinking before it leaves again (lovely image . . . .). There are plenty of possibilities other than stress. Given that you just got one of the two recently, he could have an illness or parasite of some kind that was in the incubation stage when you got him and wasn't noticable. Only now he's actively sick, and has passed it on to your other kit. I'd really call the vet in the morning and ask whether they think you should wait or bring them in. No one wants to spend limited funds unnecessarily, but better to go and find out you didn't need to than to not go and find out that you DID need to (because they've gotten REALLY sick).
Don't know what your set-up is, but don't keep the food and water near the cat box.