Ditto on the too fatty comment. Best burgers in my experience are ground sirloin or similar.
Start with good ingredients. Ditch that E36 M3ty Wal-Mart meat and hit up a butcher or decent grinds from a medium to high end grocery store. I get my ground beef from my uncle, a dairy farmer. Its primo. Very lean, and he gets everything but the steaks ground, so there is tons of flavor and not as much lip and assmeat.
You can mix in a little worstechire, ketchup, eggs, breadcrumbs, cracker crumbs, or my favorite, raw oatmeal, into your burgers but then you are pretty much making meat loaf, which is different than a hamburger. Adding eggs to already fatty beef without a substrate like bread crumbs or oatmeal is just making the problem worse.
A burger press? Really? Make a tennis ball sized ball of meat in your hands, then pat it out until its flat and even. No wonder our country is going to crap. It would not surprise me that your press is actually giving you problems, introducing air pockets and discontinuities into your patties.
I also agree with the "leave them on the grille longer before flipping" They can cook almost all the way through from one side, they are not a steak so lower heat is your friend. Like a lowrider, low and slow! Flip with a spatula or flipper, not tongs. If they show even the slightest signs of falling apart, bail, close the grille, turn down the heat, and come back in 2 minutes or so.
One of the common mistakes to grilling is over-flipping. Get to know your grille, flip once, add cheese if thats your thing, pull them off. Dont be constantly lifting the lid and flipping.
Say it with me! Good food comes from good ingredients!