The full-clause form
You are correct: how about can also take a full clause with a subject of its own. A common sort of example is “How about we eat at Sparky’s Diner?” Judging by this graph, this phrasing only started becoming common in print around 1980, and in speech it might be only about 100 years old. The gerund form with an explicit subject (see below) is older and more formal, although that appears to be changing.
In How about subject verb {objects}?, the verb is in the subjunctive mood, but usually people put it in the indicative even though the meaning is subjunctive. Here are two examples in the subjunctive mood:
How about dad pay for the car?
How about dad not pay for the car?
Notice that the negation is done in the usual way for the subjunctive: by putting not in front of the verb, without adding an auxiliary verb. In the indicative mood, these sentences would be:
How about dad pays for the car?
How about dad doesn’t pay for the car?
Here, negation works as usual for the indicative mood.
The gerund form
The gerund form can also take a subject, usually in the possessive case:
How about your taking out the trash?
How about dad’s paying for the car?
This explains the missing “us” you noticed. Made explicit (as is almost never done), that would be:
How about our not talking to her anymore?
Notice that negation works as usual for gerunds (that is, without an auxiliary verb).
The present-participle form
You can also say:
How about dad paying for the car?
How about dad not paying for the car?
How about you taking out the trash?
Notice that negation works as usual for participles (that is, without an auxiliary verb). You can even do this:
How about us not talking to her anymore?
Why all this makes sense
The above all makes sense and doesn’t come across as ungrammatical, even in the rarer forms, because how about x simply raises x for consideration or to get an answer from the listener. The x can be anything: a physical object, a fact, or an unrealized possibility.
“How about this necklace?” asks for the listener’s thoughts about the necklace in regard to whatever the current topic of interest is. Depending on context, the meaning could be equivalent to “How would you like to wear this necklace?” or “How about I wear this necklace?” or “Do you think this is a remarkable necklace?” or even “Admit that this necklace would not have been found in your suitcase if you weren’t cheating on me!”
Raising an unrealized possibility or a fact for comment is really no different than raising an object for comment. The full clause or gerund clause still functions as a noun.
If it’s an unrealized possibility, like “How about you take out the trash?” or “How about taking out the trash?”, then it’s probably a suggestion or proposal.
Subjects are optional on gerunds, so both of these are grammatical: “How about taking out the trash?” and “How about your taking out the trash?”
You can also describe a possibility by naming an object and giving it an imagined adjective such as a present participle, as in: “How about you taking out the trash?” (As a strong hint to take out the trash, this form definitely comes across as informal and disrespectful.)
If it’s an actual fact, like “How about your missing all but two games last season?”, then you can’t use the full-clause form. That would contradict the subjunctive mood inherent in the full-clause form.
I can’t think of any other situations where a full clause can be the object of about. For example, you can’t say “He’s talking about we hire John.” Maybe this is why you haven’t come across it in books or classes about English. (Most likely, books and classes skip it because it’s relatively new.)
The only form that you really can’t use is the infinitive:
How about me pay for the car?
