By definition, x ∈ ⋃F iff there exists an A ∈ F such that x ∈ A. Therefore, ⋃∅ = ∅. So, both claims above are true regardless whether F and G are empty. On the other hand, x ∈ ⋂F iff for all A ∈ F it is the case that x ∈ A. If there is an assumed universal set U, then ⋂∅ = U. However, in general ⋂∅ is not defined because there is no set of all sets. Thus, the claim ⋂(F ∪ G) ⊆ ⋂F ∪ ⋂G makes sense only for nonempty F and G.
One guess why the first statement in the OP requires nonempty sets is that the author knew that either general intersections or general unions are not defined for empty families but did not want to remember which one is not defined.