The near misses. One example per category — the rest you'll have to work out for yourself.
No land border at all. Ireland and the United Kingdom are the clearest miss: they share the island of Ireland and border each other across it, so neither qualifies. Any country that shares its island with a neighbour is out.
Exactly two colours across the whole flag. Portugal looks like green and red until you count the coat of arms, which adds three more. Emblems bring their own colours with them.
Most widely spoken, not merely official. Nigeria has English as its official language and it is not what most Nigerians speak at home. A lot of countries fail on that distinction.
Same rule. Brazil is the one people reach for by reflex, and it speaks Portuguese.
Arabic is official in more countries than it is dominant in. Chad lists it officially, and it is not the everyday language of most of the country.
No coastline on an ocean or open sea. DR Congo is the one that catches people — it has a narrow corridor to the Atlantic at the mouth of the Congo river, so it does not count.
An actual crescent, not an association. Saudi Arabia carries a sword and script, and no crescent at all.
Three or more, counted individually. One star is not enough however large it is — Vietnam has exactly one.
Measured by city proper. In most countries the capital simply is the biggest city, as in Egypt, so this square is narrower than it looks.
Exactly one land neighbour. Island countries border nobody, so they never qualify here — and countries that feel hemmed in often are not, like Spain with five.
Current member states only. Norway has never joined, despite sitting in the middle of Europe and being in the single market.
The line has to pass through the country itself, which is a narrow band. Australia sits entirely clear of it.
Each country is placed on one continent only, so a transcontinental country appears in a single square rather than two. Which one is fixed in advance and never changes between games.
Three of these are argued about often enough to be worth a longer look:
Why Ireland isn't an island country · Why Portugal's flag isn't two colours · Why Nigeria isn't an English-speaking country