By RAW, the squares do not need to be adjacent in the case of a zone but do need to be part of a valid wall. In the case of a wall, the property is the equivalent of just increasing the original size of the wall by the same number of squares.
The glossary definition of a zone:
Powers that create lingering effects that extend over an area.
And the relevant part of the "wall" definition:
A wall fills a specified number of contiguous squares within range, starting from an origin square.
The Architect's Staff adds "squares" to the "area" of a zone or wall when it is cast. Most zones are defined as the result of an area or close effect, but there is nothing in the definition of a zone that requires it to be contiguous. However, a wall has a specific definition and increasing the size of a wall has an established effect (the wall is "longer").
Note that once the wall spell is cast with the increased number of squares, those squares are part of the wall just like any others and count for purposes of the Staff's power.
All that said: I wouldn't object to a GM deciding that the resulting zone must be contiguous.