Learn Dutch: Jullie zijn erg moe en moeten uitrusten. - You are very tired and must rest.

Because it is used with the modal verb moeten (must). When a separable verb is used with a modal verb or a future auxiliary, the full, unseparated infinitive gets pushed to the very end of the clause. If there were no modal verb, you would separate it: Jullie rusten uit (You rest).

Why is there no subject after en?

Just like in English, when two parts of a sentence are joined by en (and) and they share the exact same subject, you don't have to repeat it. The subject jullie simply carries over to the second verb, moeten.

Could I use heel instead of erg to mean "very"?

Yes, absolutely! When placed before an adjective like moe (tired), both erg and heel mean "very" and are completely interchangeable.

If I wanted to add the word nu (now) to say "must rest now", where would it go?

It would go right after the modal verb: ...en moeten nu uitrusten. The modal verb moeten acts as the main conjugated verb in that clause, so other details like time (nu) come after it, while the separable infinitive uitrusten stays firmly at the very end.