Who was the first person to look at a cow (and its udders) and say “i’m gonna squeeze them and drink whatever comes out?”
It probably happened about the time our ancestors first began to domesticate cattle and other animals (something like ~10,000 years ago, IIRC). Certainly, humans would have been aware that teats provide milk, after all human females have them. It wouldn’t have been a great logical leap to realize that other mammals were similarly producing milk for their offspring which we might use for ourselves.
I think it likely someone who had seen a person milk a goat.
Direct evidence of early milking is extremely scarce. There’s nothing in the archaeological record for thousands of years after domestication began.
The oldest estimates of written records come from 4000-3000 BCE
Evidence of the Iron Age pinta
I’m not sure which evidence in particular they’re talking of, but the earliest actual representation I can find in my files is a milking scene at Al Ubaid, from the Early Dynastic Sumerian period, thought to be about 2900 BCE. It can be found in Frederick J. Simoons, “The antiquity of dairying in Asia and Africa.” Geographical Review, 61:431-439, 1971.
Quick! Goat or cow?
How much do I want to bet its a sheep. 
Cows, although Simoons also says:
You laugh, but it’s still better than I draw.
You want the guy’s name or number?
Funny a genuine question would get such a response.
Well, moriah was merely pointing out (correctly) that asking which specific person first did something along these lines is unlikely to get a definite answer. Pretty much anything that happened in prehistory is lost to the mists of times, in terms of exact details, such as this.
Whilst I understand the significance of what was said, the unwritten “tone” seemed a little uncalled for.
Ummm, you might want to read a few of Cecil’s answers. My tone was downright cloyingly pleasant in comparison. 
And considering the less than straighforward way you phrased the original question ("…I’m gonna squeeze them [udders]…"), I didn’t think I needed to throw in a smilie when taking a good-natured jab that also made a point. I’ve been the recipient of many such a jab. The usual reaction of the regulars is to say, “Well, that’s a good one on me, I must say, you rapscallion, you!”
If you need those smilies in order not to get offended, pumpkin, you may want to consider developing thicker skin before wading deeper on the board.
Peace.
I’d just like to point out that this is almost a word-for-word quote from Calvin and Hobbes.
Calvin and Hobbes rock.
Well, step up and answer the OP then. Was it Calvin or Hobbes who first said it?
(I first heard it myself from some redneck comedian circa 1985.)