"curvilinear" could mean anything geometrically not a straight line on the scale being used. So, that could mean many things, including behaviour best tackled with powers of another variable, exponentials, logarithms, trigonometric and hyperbolic functions, etc., etc.
Using logistic regression does not change what is standard in any kind of regression-like modelling: You can have whatever predictors (so-called independent variables) in your model that make sense, so long as there are sufficient data.
Those general statements aside, trying a quadratic term in your model as well as a linear term is often a good simple way of adding some curvature. Because you are using a logit scale, intuition needs refining here. In particular, if your coefficient on the squared term is negative, you are fitting a kind of bell shape on the probability scale. This is often a feature in e.g. ecology where probability of occurrence of organisms is greatest for some intermediate value of an environmental predictor. In simple terms, it can be too hot, about right, too cold, and so forth. See http://www.cambridge.org/gb/knowledge/isbn/item5708032/ for one good account.
I trust that others will add advice about SPSS.