Sounds like just a bout of food poisoning due to the under-cooked mince. Should be out of your system in 24-48 and a bit hours. If not go see a doctor, keep your fluids up and ya stay away from dairy.
Minced meat has way too many surface areas that when cooked in a patty style if not done propperly you get sick. Most meat pattys you get from fast food places are pre-cooked/blast chilled(frozen) then re-cooked so your fine.
Steaks only have outside tissue areas that need to be sealed, as steak muscle is pure, hence why you can eat blue/rare cooked meat and not get sick.
-N