Are you required to place shields on your ships during setup? Could I choose to forgo shields on one of my ships?

Edit: Ah well, it wouldn't have been competitive anyway, just a silly thought for a list in a moment of being half-awake.

Yes. No.

Rules Reference Guide. Page 17. Setup. Item 6. Place a number of shield tokens on each Ship card equal to that card’s shield value.

When all else fails, and sometimes before, read the rules.