DES/AES invertibility
DES is a Feistel cipher where the round function doesn't need to be invertible. Therefore, when designing a Feistel cipher you will have invertible and non-invertible S-box options.

In contrast, AES is a substitution-permutation network where the structure is completely different. If AES operations are not invertible, you will not be able to decrypt anything. So, the second claim is wrong.