I agree with our friend from Bahrain; the way I phrase it for my students (whose proficiency levels vary quite widely) is that a prime number has exactly two DISTINCT factors - i.e., 1 is not prime because 1 and itself are not DISTINCT. I usually explain this early on, and then have students give me examples of prime numbers. I've taught several hundred students, and I've never encountered a lingering "!???" regarding prime numbers. (In all fairness, though, I'm a Math guy, and I like your explanations too. If someone plans to mess with our theorems, they'd better have a darn good reason!!)
Regarding (-a)(-b)=ab, I tell them that a negative is an opposite - e.g., for temperature, 50 degrees is 50 degrees from zero one way, while -50 degrees is 50 away from zero in the opposite way), and that the negatives cancel each other out, because the opposite of the opposite will be the thing you started with. (A visual, such as a number line or a thermometer, can be very helpful here.) Honestly, this works s fair number of times, but certainly not always. I believe that, in the same way that we mathematicians don't like people trying to disrupt our theorems, many non-mathematicians simply don't care for negative numbers!