NO! No never ever ever.
Squish, flat burger, bye-bye flavor...
Though working in some restaurants you need that burger to cook to well in 7-8 minutes, that's when you put a cast iron skillet or press on it, not my rules, just trying to get the food out.
i usually press down just a little. not to squeeze out any fat, but to tighten up the burger a bit. it also depends on how much fat i'm starting with. if i'm using a 15 to 20% fat grind, and i don't press a couple times, the burger ends up too loose. grinds with 5 to 10% tend to hold together better, and mashing doesn't seem nessesary.
Not me. Most people get tempted to do that because they make a fair size burger only to see it contract and start to look like a meatball. So they then try to flatten it out and squeeze the juices out of it. Instead of doing that take the hamburger and before you put it on the grill, after making your pattie, put a good thumb print dimple in the middle of the burger. Then place it on the grill. You'll find it won't swell up to meat ball size again.
I don't even cook beef burgers... we do turkey burgers with a particular sauce (don't remember the name) and chopped onions. They taste better than any beef burger I've had in many years. Or we do lamb burgers - which are delicious.