What would be the Sref i.e. Reference Wing Area for Drag Build Up Calculation in case of a box wing design?
You could argue for one or for both, but ultimately, it doesn't matter - as long as you are consistent and use the same reference area throughout your calculation/analysis.
If I personally were to choose, I would go for both, on the basis of total lifting area.
EDIT for details ------
Well yet again, it doesn't matter. You will get some value for your $C_{Dmin}$. Then later for other calculations you will use the same $S_{ref}$ and you will get comparable drag coefficients. Finally, when you use $D_{total} \text{ [N]} = C_{D} \cdot 0.5 \cdot \rho \cdot v^2 \cdot S_{ref}$ you will input same $S_{ref}$ and you will get the same Newtons regardless of chosen $S_{ref}$.
What matters is that you're consistent, i.e. if you use $S_{ref} = 10 \text{ m}^2$, use that for everything. If you use $S_{ref} = 20 \text{ m}^2$, use that for everything.
Additionally, do recheck the wetted area argument. $S_{wet} \neq S_{ref}$, because $S_{wet}$ includes both upper and lower surface of each wing, whereas $S_{ref}$ includes only the planform surface.