I suggest you look at Mather's "Hurewicz Theorems for Pairs and Squares" in which Blakers-Massey is derived from his   Cube Theorems.  It works for all spaces, but the maps involved
have to be $m$- and $n$-connected with $m,n \geq 2$.

The basic inputs to the cube theorems are: (1)  Hurewicz/Dold local-to-global criteria for fibrations (or weak fibrations) and (2) the pullback of a cofibration by a fibration is a cofibration.

(Actually, (2) is not used in this proof.)

ADDING ON:  The proof is extremely cute and easy.  

Given a homotopy pushout square 
(spaces $A,B,C$ and $D$) pull back from the path fibration $\mathcal{P}(D) \to D$
to get another square.  The Second Cube Theorem tells you it is a homotopy pushout, 
and comparing connectivities of maps shows that it suffices to prove the B-M theorem
for the new square.

But the new square is a square of simply-connected spaces (since the spaces involved
are fibers of the maps involved), and the homotopy pushout is contractible.  Now the comparison map to the homotopy pullback may be identified (after suspension) with the inclusion of the wedge into the product, and that connectivity is easy to determine.