Block version of Maurey Pisier theorem

I believe that the answer is NO. I do not know a counterexample, however, in positive direction one can do the following.

If the basis is unconditional you can define type/cotype on disjointly supported vectors. Then the corresponding result is true. This is written here (with slightly different language), see Theorem 5.6.

The reference that you are looking for must be the paper by Milman and Sharir. A warning though, what they call block version is actually disjointly supported version (not successive blocks) i mentioned above.

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Fa.Functional Analysis

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