How do you translate “Freedom means choosing your burden.” (in the correct grammar) in Greek language? And in Greek writing as well?

Tough, because it’s got to be idiomatic.

In Modern Greek: You have the choice of the formal ελευθερία or the colloquial λευτεριά. We’ll go colloquial, it has more resonances.
Burden: φορτίο is formal. Γομάρι is colloquial, but it also has ended up meaning beast of burden, hence jackass, so it won’t do. Unless you have religious objections, the culturally appropriate equivalent is probably “your cross” (to carry). So:

Λευτεριά είναι να διαλέγεις το σταυρό σου.

or maybe, spelling out “to carry”:

Λευτεριά είναι να διαλέγεις το σταυρό που θα κουβαλάς.

Yes, everything is longer in Greek.