As the other's have already pointed out, that's a sentence indicating a strong aversion to an object, Like "I wouldn't even take that if you'd give it to me for free". It's not necessarily an aversion to the person or getting presents from them and you wouldn't say that if you'd actually had received it as a genuine present.
It's more of a statement that you could apply when someone asks you for the value of an object, like if they plan to buy something or are throwing out stuff they have around the house and you argue that it's completely worthless.
In terms of the "wollen", that just means "to want sth/sb". And in that case it's used in subjunctive ("Konjunktiv") which is a mode in German implying some possibility or conditional statement. Like no one is actually offering that object as a gift, but "if they would, then they would reject it". So they argue that "They wouldn't even want to have it if someone would offer it to them as a gift".
Er will es nicht gewusst, gesehen haben (= behauptet, es nicht
gewusst, gesehen zu haben)
That construct of speech is a little more tricky. A possible scenario is something like this:
Person A claims: "I didn't know or see anything!"
Now Person B reporting to Person C could just say:
Person B to Person C: He claims to have no known or seen anything.
But Person B is suspicious that Person A might be lying and to communicate that suspicion he doesn't say that Person A claims that but instead says something along the lines of:
"Person A would like to have not known or seen anything" or "Person A wants me to believe that he hasn't known or seen anything."
So the "wollen" does not mean "claim to have done something" it's rather implying that someone would have liked to have done/not done something or would want others to believe (that he did) sth. So it's still somewhat implying a desire/sth that you want.