orbitals - Do electrons only fill 'spin up' first? Or could it start filling 'down spins' first?

Due to Hund's rule, electrons start filling up the orbitals without pairing up. When this is happening, do the electrons all fill up the 'up' spin? Could they fill in the 'down' spin? Why do they prioritize the 'up' spin if they do?

Answer

No it does not matter whether you start with 'spin up' or 'spin down' electrons as long as you are consistent: If you start filling some orbitals with 'spin up' electrons you will have to keep doing this for all the orbitals you want to fill.

But: The convention is that you start with 'spin up' as this usually is taken to represent the $\frac{1}{2} \! \hbar$ eigenvalue of the $z$ component of the spin angular momentum operator and you should adapt to that in order to avoid confusion.