You are correct, a new speaker is capitalized. It's a new sentence, and the first letter of a new sentence is always capitalized. Period.
When you say it is continuing the same sentence - it's a sticky situation, but it isn't, grammatically. You need to think of everything within quotes as a single part of the sentence, just like nouns, verbs, or adjectives. The rules of grammar can be ignored within a quote (because people don't strictly adhere to grammar when speaking). That's why a quote can pick up in the middle of a sentence. That doesn't mean it, as part of what you are reading, is still the previous sentence.
One thing I would like to note on the second line:
His eyes widened, “I can’t do that! Her face—”
The comma there should be a period. I see lots of people get confused about this. The quotation is just another of the sentence. 'His eyes widened' is a complete sentence. The quotation isn't a part of it; it's a whole separate sentence. Therefore the comma should be a period, and the first letter of the quote would be capitalized (which it already is because it is 'I' in this case).
Now if you wanted the comma there, you would write something like:
His eyes widening, he said, "I can't do that! Her face—”
If you realize that the quote is just a phrase within the sentence, it's easy to see why you use a comma there. In this case, it is the direct object of the sentence. It is what was said.