Learn Polish: Ten wymagający szef nie akceptuje byle czego. - This demanding boss does not accept just anything.

Byle co is the base form. However, this sentence uses a negated verb: nie akceptuje. In Polish, negating a verb that normally takes the accusative case forces the direct object into the genitive case (this is the "genitive of negation"). Therefore, co changes to its genitive form, czego.

Could I use czegokolwiek instead of byle czego here?

You could, but it changes the nuance. Czegokolwiek comes from cokolwiek (anything at all), which is neutral. Byle co and its forms carry a negative, dismissive tone, meaning "any old thing" or "something of low quality." By using byle czego, the sentence emphasizes that the boss has high standards and won't accept mediocre work.

Does the byle part ever change when we decline these pronouns?

No, the prefix byle is completely unchangeable. When you need a different case, you only decline the pronoun attached to it. For example, kto becomes kogo (byle kogo), and co becomes czemu in the dative case (byle czemu).

Wymagający looks like a verb. What form is this?