IIRC, they explained the Quarian immune system as follows:
The Quarians evolved on a planet whose biosphere developed around cooperation rather than competition. Instead of developing immune systems that fight off other creatures, all the lifeforms on the planet developed systems that adapted to each other, presumably for a symbiotic mutually beneficial relationship. After generations in the migrant fleet, their "adaptive" immune systems adapted to the conditions of the Migrant Fleet, and became permanently adapted over time. Now they can't even return to their homeworld's atmosphere without generations of corrective genetic engineering.
They don't actually lack an immune system (like someone with AIDS), it's just that their alternative system malfunctions to the point that encountering foreign substances creates a kind of "allergic" reaction that can show in different forms of sneezing, coughing, nausea, vomiting, etc. That's why they get a generic kind of "sickness" instead of a severe manifestation of whatever microbe they have actually encountered. AIDS probably sucks more than having a Quarian's immune system. I'd rather die of the sniffles than from being swarmed under from every kind of horrible disease at the same time.