Hi there Serena,
Welcome to Permies!
Yes, growing plants is a wonderful thing to do - good on you not to let anything stop you!
It would be a good idea to try and grow them out in the patio. Squash loves a sunny, warm location, rich soil and regular watering.
When you move your seedlings outside from the apartment, be mindful with letting them get used to the UV rays of the sun gradually.
My method is to do the moving-out on a cloudy day and also to put them in a spot (still in their original pot) where the sunlight doesn't get to them all day long.
After 2-3 days I transplant the seedlings to their final spot. It's best to transplant your seedling with the whole soil clump that comes from the seedling pot.
Water them before and after transplanting and then leave to recover for 3-5 days. After that water them if they look thirsty (or stick your finger into the soil to feel if it's dry) .
It's normal that the plants look sad after transplanting. They get stress from such a change. They usually perk up again within 1-2 days.
A huge pot or a grow box should work well for your spaghetti squash. Keep in mind that the plants need about 1m2 (one square meter) of space per plant.
Some squashes are vining, for those trellising would work well. Others grow from a central point like a rosette.
Harvest your squashes when they're small to medium in size. If you let the fruit grow really large, the plant starts producing seeds and will put all its energy to making that one fruit with seeds (stops producing more fruit).
It's actually a good idea to let one plant (or a couple) produce such a seed-producing squash. Save seeds!
Find out more about saving your own seeds and why it's such a good idea
here.
Good luck!