We've built from both straw and hay.
We only used hay as we grow this and buy straw. so cheaper to use hay for us, but as HH says they do slowly eat it, so straw is preference, but if you have old hay, you can use.
Build like bricks, ie overlap each row, and tie every bale to all it's neighbours above below and each side.
Some vertical stakes will make it stonger, particularly at entrance.
Tin or tarpaulin roof.
We have a de-luxe model (home for 5-7 full size sows) which has low boarding on outside and inside, which is now 4 years old. Needs an annual mot (add new bales where they get chewed), but otherwise is one of the pigs favorites arks.