I can't believe they'd avoid going to medical - they might grumble about it, but you're right, they need their bodies in the best condition possible.
Psychologists, on the other hand... that's harder to tell. For Clint, for whatever reason, I can't imagine that he likes them, but on the other hand, they do help. In Nat's case, though, she's got a history (as far as I can tell) of keeping her cards close to her chest. I can't imagine she'd like the idea of opening up to someone who will write everything down to go into a file...