I breastfed DD exclusively until 2 months, then mix fed for a further month, then from 3 months she went onto just formula.
By contrast DS1 and DS2 were exclusively breastfed until a year and a half. (well, DS1 for about 19 months and DS2 for 18 months).
Out of the three of them, DS2 is probably the most prone to catching stuff. He has mild asthma and needs inhalers, and gets chest infections the most often. If there are any colds going, he gets them worst of the three kids.
Next after that is DD. She is the most likely to get a tummy bug and throw up if there is one going around. If she has a cold it lingers a bit, but not as much as with DS2.
DS1 is the healthiest of the lot. No idea why. He was the smallest at birth (I say smallest, but he was still big, 10lbs 1oz) out of the three of them, but then again he was breastfed the longest (though only marginally). He was also the one who had the most issues with eating solids when he was a baby. He refused solids altogether until about 15 months old, and his weight dropped to 9th percentile from 91st. He was essentially just surviving on breastmilk!
If he gets poorly, he gets over it the quickest. Colds don't tend to linger with him, and tummy bugs are over the quickest.
When the three of them caught chicken pox last year, DS1 got it first and he had it the least severe. Then DD and DS2 got it, and out of the two of them, DS2 had it the worst, though only marginally.
So, no, there doesn't seem to be any correlation between the breast feeding and how ill my kids have been. I can't really make any conclusions to be honest.