Exploratory Factor Analysis gets the 'exploratory' out of the fact that you do not specify in advance which factor loads on which items. All factors are allowed to load on all items. You always, always need to specify the number of factors because, if you do not, then your model is not identified and you cannot obtain parameter estimates.

Whenever you don't ask SPSS to select the number of factors, SPSS uses a flawed, problematic rule that was popular in the last century where only eigenvalues greater than 1 are selected. But whether you formally choose the number of factors or you let SPSS do it, the number of factors always has to be specified in advance.

In my opinion, one as a researcher should always select the number of factors. There is a wide variety of 'pointers' to probe the number of factors (Parallel Analysis, Velicer's Minimum Average Partial correlations, etc.) which you can use to inform your judgement, but nothing will ever trump the theory that substantiates the purpose behind your analysis. Latent variable models in psychometrics are mostly theory-driven and it is important not to rely on heuristics or quick-n-easy solutions to fundamental problems of research.