When you squeeze something, you put pressure on it. If you squeeze your toothpaste from the bottom up, you'll get the most out of your tube.
The word squeeze has to do with force or pressure. When you squeeze something, like an orange, you are crushing it to extract juice. When you squeeze into a pair of jeans, you're forcing yourself into them. When you give someone a squeeze, you're giving them a tight hug. And when you're in a squeeze, you're in a tight spot, or a bind.