Did English ever have a word for 'yes' for negative questions?

The Germans have doch and the French have si as a word that means "yes" in response to a negative question, such as:

Don't you want some ice-cream?
Yes [I do]!

In English, we only have yes (as far as I know) and further clarification is required in order to be unambiguous.

Did we ever have such a word in English and, if so, what happened to it? If not, given the French and Germanic influences on English, any idea why not?

Such words existed in Early Modern English (roughly 1450–1650), and they were… yes and no. However, the answers to positive questions at the time were yea and nay. You could summarize their use as such:

• Will he not go? — Yes, he will.
• Will he not go? — No, he will not.
• Will he go? — Yea, he will.
• Will he go? — Nay, he will not.

It is well detailed in this Wikipedia page (which I first read in an answer by z7sg earlier today):

Whilst Modern English has a two-form system for affirmatives and negatives, Early Modern English in fact had a four-form system, comprising the words yea, nay, yes, and no. [...] The answers to positively framed questions ("Will he go?") were yea and nay, whilst the answers to negatively framed questions ("Will he not go?") were yes and no. This subtle grammatical nicety of Early Modern English is recorded by Sir Thomas More in his critique of William Tyndale's translation of the New Testament into Early Modern English.

^{As you noted Modern French, as other languages, has a three-form system where both negative answers are the same (oui is the affirmative answer to a positive question, si is the affirmative answer to a negative question, and non is the negative answer to any type of question).}