Learn Portuguese (Portugal): Ambos os cães estão doentes.

In European Portuguese, the quantifier ambos (both) must be followed by a definite article when it comes directly before a noun. Because cães is a masculine plural noun, we need the masculine plural article os. You cannot translate the English phrase word-for-word; it must be ambos os cães.

If we already know we are talking about the dogs, how would we just say "Both are sick"?

When ambos acts as a pronoun standing in for the noun, you drop the article. You would simply say Ambos estão doentes. The article os is only mandatory when the noun explicitly follows the quantifier.

The plural of cão is cães. Is there a rule for why it doesn't end in -ãos or -ões?

Words ending in -ão can form their plurals in three ways: -ões (the most common), -ãos, or -ães. Unfortunately, there is no reliable rule to predict which one a word will take. The word cão belongs to the smaller group that takes -ães. At an intermediate level, these exceptions simply have to be memorized.