Part of the problem is that GHC's error messages have to account for a lot of complex typing extensions you aren't using, so they aren't clear. I'll try to explain what's going wrong.

If you look at the function,

area (SquareConstructor side) = side * side

in isolation (that is, not as part of the class instance), it has the type "forall a. Num a => SquareConstructor a -> a".

The function in the class declaration has type "forall a b. (ShapeInterface a, Num b) => a -> b". The problem is that a and b are independent variables, but the instance declaration for SquareType a requires them to be related.

I'm not sure which way you were trying to parameterize things, but here are some possibilities:

(1) If every shape type is parameteric, then you can make ShapeInterface a class of type constructors.

    class ShapeInterface shape where
        area :: Num a => shape a -> a

    instance ShapeInterface SquareType where
        area (SquareConstructor side) = side * side

(2) If only some shape types are parametric, you can use a multi-parameter type class to express a relationship between the shape type and the area type:

    class ShapeInterface shape area where
        area :: shape -> area

    instance (Num a) => ShapeInterface (SquareType a) a where
        area (SquareConstructor side) = side * side

(3) If you only need to be parameteric over some subset of numeric types, you can use conversion functions:

    class ShapeIterface shape where
        area :: shape -> Rational

    class (Real a) => ShapeInterface (SquareType a) where
        area (SquareConstructor side) = toRational (side * side)

(Equivalently, you can replace Rational and Real with Integer and Integral.)