Can an egg be forced into a bottle by lowering the pressure inside the bottle using cold air? I just asked a question about why ice-cold water inside a thermos results in what feels like suction on the cap. The answer stated that the cold water cools the air inside the thermos, thus slowing it down, thereby decreasing the pressure on the inside of the cap relative to the high atmospheric pressure pushing on it from the outside. 
But what about the egg and bottle demonstration in which a hard-boiled egg is forced into a bottle with atmospheric pressure by lowering the pressure inside the bottle? In that demonstration, the air pressure is lowered by heating the air inside the bottle. Could the same effect be achieved by somehow rapidly cooling the air inside the bottle? Or if not a bottle, then a thermos? 
 A: 
the air [pressure] is lowered by heating the air inside the bottle.

I think you need to watch the video again.  The air pressure is raised by heating the air, not lowered.  This increase in pressure forces air out of the bottle, which is visible by the egg vibrating on the rim.  The egg acts like a one-way valve.  When the pressure inside is greater, it pushes the egg up, but only slightly.  Once a gap appears, the air can rush by.
After the flame goes out, the air inside cools, lowering the pressure just like in the thermos.  This time the egg cannot lift slightly to equalize the pressure and is instead pushed into the bottle by the unbalanced forces.
A: I like BowlofRed's answer, but I think there's more going on here than that.  I also don't believe that straight forward cooling would suck in the egg that fast.   The egg gets sucked in very quickly.
Certainly as the burning starts, air is escaping.   It's impossible to see precisely when the egg stops vibrating, as some vibration might be quite small and hard to see but what visibly happens around 1:18 - 1:19 is the egg begins to get sucked down and as this happens, smoke begins to fill the bottle.
Newsprint is primarily Cellulose:  Source and Cellulose has a chemical structure of $C_6H_{10}O_5$.
So, the simplified chemistry of burning newspaper is $C_6H_{10}O_5 + 6 O_2$ gives us $6CO_2 + 5H_20$, so you're replacing 6 Oxygen molecules with $6 CO_2$ and $5 H_20$ (in gaseous form), so it's not just the temperature but 11 molecules of gas are replacing 6 and that also causes air to rush out and vibrate the egg.
The flame itself is also quite hot (Source) and this not only heats of the inside of the bottle, but there is a fair bit of localized displacement by the flame and the fast moving molecules from the chemical reaction.   As the flame begins to shrink, in part due to lack of Oxygen the smaller flame creates less displacement and this creates the suction, even though the flame is still burning some as the egg begins to get sucked in and the fire is still adding gas molecules and generating some heat, but the reduced displacement when the flame shrinks begins to suck the egg in.   Now it might be semantics to call it "displacement" vs heat.   I see that, but the important point is that as the flame shrinks the egg only gets sucked in a little, but in a reasonably closed system like that even a small flame should continue to add some heat, so there's more at play than just temperature.
The 2nd thing, which is visibly obvious is that smoke quickly fills the inside.  This is because the gaseous $H_20$ is much too concentrated for the average temperature inside the bottle and it quickly forms into tiny suspended droplets of water and this gas into tiny droplets has a pretty significant reduction the air pressure, enough to suck the egg in in 2 seconds.  Condensation is an exothermic process, so this water vapor to visible droplets of smoke/steam should warm the bottle further - a little bit, but the effect of the reduced air pressure due to condensation is a greater factor than rising temperature or anything else at that point.  
The abundance of the water vapor is basically super saturated and as it becomes visible, the bottle is losing air pressure.  Cooling wouldn't happen that fast.  For example, if you had a warm bottle like that one with an egg snug on top and you put it in the fridge, it would (I think) take quite a while longer than 2 seconds to suck the egg in.  
Now if you had a different fuel, say pure aluminum (which if you got it burning it would melt the glass bottle), but anyway, Aluminum sucks Oxygen out of the air without giving anything back.  That kind of reaction might suck the egg straight in without any vibration from escaped gas at all.
And if you had Gunpowder, which generates a huge amount of gas when it burns, the egg would be shot high into the air and the bottle, well, lets just say, do this one behind a protective screen.  The chemical changes in that experiment can't be ignored, so it really can't be compared to the thermos.
