Do kids get sick more often during the terrible twos?

No. Like adults, kids get sick more often, the more often they are around others who are sick. They may be in daycare with 3+ nonsibling kids, 6+ hours a day, 4+ days a week; or siblings or parents may be around more people now than when the child was younger. Just being a certain age isn't the key risk factor.
No. As far as i know there is no evidence to support any notable increase in 'sick' children(viral or bacterial infections) during the "terrible two's". This period of increased misbehaving is actually normal as the young child begins to test his or her boundaries; as well as the parents' patience. Just continuing to be understanding, reassuring, and supportive will get you through.