I am planning for a Ultra distance (~230 km) cycling. I don't own a good bicycle for that and when I was checking rental bicycles (Road bike & cross bike) one of the shopkeeper suggested me to take cross bike saying that rate of the risk flat tire of Road bike is more than of Cross bike because of the thin tire. I didn't understand the logic. Could anyone explain the logic behind this?
Excluding tires so old and worn they obviously need replacement, there are two main root causes of punctures in well maintained bicycles.
Low Pressure causing 'snake bite' punctures. When the pressure is too low, and you hit a bump, the tire can pinch the tube against the rim and cause a puncture. Small tires have to run at higher pressure to stop this happening and have lower volume. All tires leak air slowly - so the combination of low volume and higher pressure means they need to be pumped up more often than larger tires - in some cases weekly.
Additionally a slow leak that may go unnoticed for longer in a larger tire will become catastrophic to a low volume tire much quicker.
Punctures where an object penetrates the tire is what most people think of. While small tires have a smaller area to hit such an object, they run higher pressure and are therefore harder - which generates more force to push the object into the tire. Generally a larger tire spreads the loads over a larger area, so less likely to push an object into the tire. Another aspect is something sharp like glass may cut a tire and the higher pressure of the smaller tire will push the tube though the cut.
The differences are small, and it depends as much on tire and wheel choice as size, but all else being equal, experience shows that most cyclists puncture less on larger tires.
As far as punctures go, you need to be able to deal with them on a long ride regardless of the bike you are riding. If the road surface is good quality and generally swept clean, then a road bike is more suited and practically no more likely to puncture than a cross bike. If the road potholed and rough, with lots of stones and litter then larger tires are better choice. Larger tires also offer a more comfortable ride and for this reason may be more suited for you. The rolling and aerodynamic efficiencies of narrow tires may make a difference to a pro rider trying to break away from the peloton, but you may find the few seconds per hour difference in speed less enticing at the end of 200+km or riding.
Also I suggest you take the bike out for a decent ride before the event. 230km on a bike you have ridden no further than around a car park is a big risk.
Without seeing the exact tires that are on the bike, we need to speculate. Mattnz discusses how the width across the tire affects the chances of puncturing. Another factor is the thickness of the tire carcass, which might be what the shopkeeper meant. A thin-wall tire will obviously be more prone to punctures than a thick-wall tire, at any given tire width.
The cross bike might be set up with knobby tires (since that's what you'd use for cyclocross racing). The tire lugs give you a buffer between the tire carcass and the ground, which would improve puncture resistance, but they'll roll much less efficiently on hard pavement, and would not be a good choice for a 230-km ride.
As someone who routinely rides that sort of distance, I'd say the biggest worry about punctures is knowing how to fix them, and being equipped (a couple of tubes, patch kit, pump, tyre levers). They're rare even on the poorly maintained back roads I often find myself on, and can happen whatever tyres you ride.
Over the last two years, in 26 rides of over 200km, I've had one pinch flat, and a recurrent slow puncture caused by one tyre that was worn out on the inside. That was on marathon supremes and duranos, which are protected against punctures; riders with less tough tyres get more punctures but still not all that many. Comfort is more of a concern over that distance; I've worked my way down from 35mm to 28mm on the front, and that's a little harsh for my wrists.
The cross bike may well have a more relaxed geometry, for which your body is likely to thank you, especially if you're not used to a race-like posture. That's why I ride a tourer.
You could always buy a set of wide road tyres for the cross bike and keep them for future rides. Buy them through the hire shop and I'd expect them to swap them on and off for you. If you mention this idea, they may offer an alternative.