You cannot use nominative in this example - Henry and I are not opinions
Henry und ich sind Lehrer.
Would be using nominative, just as in English.
Just as well, verschiedener Meinung is not dative in your example case.
Let's consider another example to make it clearer
Der Katholik und der Buddhist sind unterschiedlichen Glaubens
We're using a masculine substantive to be able to detect the case properly - Glaubens is clearly genitive.
This construct is called genitivus qualitatis or, in German, Genitiv der Beschaffenheit and is used to describe a qualitative attribute of an object. Other examples of this usage are
Ich bin protestantischen Glaubens
die Freude war kurzer Dauer
der Mann war mittleren Alters
Ihre Familie war chinesischen Ursprungs
Modern German tends to use this specific case less and less and replaces it with prepositional expressions like
der Mann war von mittlerem Alter
die Freude war von kurzer Dauer
(Here, von asks for the dative)