Generally, “squeeze” is used for objects that you can apply pressure around or apply pressure on opposite sides of. Think of it as two forces trying to meet at the center of the object. For example, you can squeeze a person tightly if you give them a hug. You can squeeze your fist together. You can squeeze a lemon slice. You can squeeze a bottle of sauce. “Press” is used to describe one force going in one direction, like pressing a button, pressing a sticker on paper, pressing clothes with an iron.

Both words have a number of meanings and can be different parts of speech and can be used figuratively. ‘Squeezing’ suggests movement. For example, we say ‘squeeze the trigger of a gun’ (also ‘pull’). The trigger moves. Using ‘squeeze’ intransitively we’d say ‘let me squeeze in ‘ (the space is too small and I’m going to have to make myself smaller to fit) Figuratively, ‘press’ is much more common. They’re pressing me to /buy an update/join/tell my story/. (Encouraging, applying pressure, but a reasonable amount) Figuratively, squeeze might be used in business and it’s much stronger. I’m being squeezed by the competition/mob. (There’s strong pressure from all sides and it is affecting me)