It is not clear what syntax to use to distinguish variables (that specify the dimensions of the space that the feasible region lives in) and parameters (basically, constants that are specified externally).

Maybe

S = @constraint(c, 1 <= (x - c)^2 <= 3 )

The last argument is the constraint; the others specify that those variables are parameters.
Then S(1) should be the constraint specialised with c = 1 etc.