negation - "He needn't say" / "He doesn't need to say"

When I studied English at school (in Italy, more than 35 years ago), we were taught a rule according to which one should say, e. g., "He needn't (need not) say", but NOT "He doesn't need to say".

If I'm not mistaken, it was explained to us that the rule holds good with the following limitations:

1) only in the negative form: thus, "He needn't say", but "He needs to say", and "Does he need to say?";

2) only in the present tense: thus, "He needn't say", but "He didn't (won't, wouldn't, etc.) need to say";

3) only when an infinitive follows: thus, "He needn't see a doctor", but "He doesn't need a doctor".

But, if I google "doesn't need to", I find 125,000,000 occurrences.

I was wondering: does the rule hold good more for British than for American English?

Also, perhaps it isn't an absolute rule, and both constructions ("He needn't say" and "He doesn't need to say") are possible.

Is the rule at least a good description of the circumstances when we CAN (but needn't necessarily) use this construction?

Thank you very much, for this and the other answers.