Learn Turkish: On tane yumurta yok. - There aren't ten eggs.

In Turkish, whenever you use a number before a noun to state how many there are, the noun must stay in its singular form. You never add the plural suffix (-lar or -ler) after a number. Because of this rule, it is always on yumurta, never on yumurtalar.

What is the word tane doing in this sentence?

Tane literally means 'piece' or 'item'. In Turkish, it acts as a generic counting word used between a number and a noun. Saying on tane yumurta is like saying 'ten pieces of egg' in English, though it just translates naturally as 'ten eggs'.

Do I absolutely need to use tane here, or can I just say On yumurta yok?

You can easily leave it out. On yumurta yok is perfectly correct grammar and means the exact same thing. However, using tane is incredibly common in spoken Turkish when counting physical objects, so it is important to get used to hearing and using it.

Why do we use yok for 'there aren't' instead of değil (not)?

Yok specifically means 'does not exist' or 'there isn't/aren't'. You use it when talking about the physical presence or absence of something (like whether there are eggs in the fridge). Değil, on the other hand, is used to negate facts or descriptions, like saying 'The egg is not big' (Yumurta büyük değil).