I am not going to be able to answer a question about a specific output of a specific routine in SPSS, but I can tell you this about multiple regression in general: yes, it correctly accounts for multi-colinearity effects.
A two-factor logistic regression assumes
In your example, x_1 might be gestational age and x_2 birthweight. The output of the logistic regression should be best-fit values for a, b_1, and b_2, along with a covariance matrix between them (and from that covariance matrix you can derive error bars on the three parameters).
Suppose gestational age and birthweight are highly correlated, and both are inversely correlated with the incidence of this pathology. This if you did a single-factor logistic regression on either x_1 or x_2, you would expect the b-coefficent to be significantly negative in either case.
Now suppose, furthermore, that, if you control for gestational age, then low birthweight actually protects against the pathology. That is, for a given gestational age, you are actually less likely to get the pathology if your birthweight is low. We just didn't see that effect in our single-factor regression on x_2, because so many of the low-birthweight babies also had low gestational age, and the pure gestational age effect swamped the pure birthweight effect. Then the two-factor logistic regression will correctly see this effect: the b_1-coefficient will be negative, but the b_2-coefficient will be positive.
The one thing that confuses me about your question is that you pose it in terms of "correlation". The regression coefficient is related to correlation in the single-factor case, but not in the multi-factor case. The corrleation between birthweight and this pathology is what it is, and looking at any number of additional factors will never change that number or "correct" it for multi-colinearity effects. But the coefficients in a multiple regression analysis will change as you add information on additional factors, in a way that correctly accounts for multi-colinearity effects that were hidden with the additonal factors were not observed.