He's right. 'Looks' functions as a linking verb (similarly as the verb 'to be') in the example above and 'well' functions as a predicate adjective. He could have also used 'good' as the predicate adjective.

However, he could have also meant 'well' to be an adverb with 'looks' functioning no longer as a linking verb, but as an intransitive verb in a different context:

1st speaker: "Only those who look well to find lost objects will find them."

2nd speaker: "Mrs. Klein has lost something in the parlor."

3rd speaker: "Oh, she'll find it!"

4th speaker, "Certainly. Mrs. Klein looks well."

In that context, 'well' functions as an adverb; in the context you cited, 'well' functions as an adjective.