Regardless of the kind of closed surface (sphere, cube, etc.) the net flux across the surface is zero unless there is net charge enclosed by the closed surface, in which case the net flux is the charge enclosed divided by the permittivity of the space according to Gauss' Law.
If there is a non-zero net flux for your sphere, it must be because there is charge enclosed. Placing it in externally generated electric field alone would not produce net flux.
Wouldn't we take the projected area of the sphere (i.e. a circle), for
calculating the electric flux? –
If you are only interested in the flux entering the sphere from an externally generated uniform electric field then of course that will be the flux across the projected surface area of the sphere ($πR^2$). But that is not the net flux across the total surface of the sphere as it will also exit the other projected surface area of the sphere. The net flux is zero over the entire surface of the sphere unless there is net charge enclosed.
Hope this helps.