The none utility (the "not a possibility" category from the Screener section) is estimated as a part-worth utility parameter during the utility estimation.
We stack the three sections of the ACBC data into one big regression matrix, with the BYO tasks, Screener tasks, and Choice Tournament tasks all being sections (series of rows) within that matrix. The None parameter is a dummy-coded (independent variable) column in that regression matrix, and it only can take on values of 1 for the Screener rows of that matrix (when the alternative being coded is the "not a possibility" alternative); for the BYO and Choice Tournament rows, that dummy coded independent variable is always 0.
Design matrix refers to the coding of the independent variables for the rows (cases) shown to the respondent, often referred to as the X matrix.