Lets say I eat 4 boiled eggs one morning and that equals up to 350 calories and I go to the gym an hour or two later on a machine and burn 350 calories there, did I burn what I just ate for breakfast?

It means you just burned the same number of calories that you ate for breakfast - depends on who you ask if the answer is that you burned what you just ate or if you burned stored calories. At least I got different answers to the same question. I tend to just say yes, you burned what you just ate for breakfast.

Beware of the 'calories burned' monitors on equipment at the gym - they are notorious for overestimating the number. I wear a heart rate monitor that shows calories burned, and sometimes wonder if even that overestimates a little.