I've had nothing but trouble with bugs in my veggies whenever I've used any kind of mulch: it provides them with too good a living environment and especially a good haven in the winter. A friend of mine who owns a CSA organic farm says he's done with mulch for the same reason. I've used both straw and wood chips. I would do whatever is easiest to remove the straw. Tilling it in would be OK, but that sounds like more work than removing and composting it.

I've read the opposite - that straw is good under squash plants. It deters the cucumber beetle from laying eggs in the soil at the base.
If you are not bothered by cucumber beetles your best bet to deter squash bugs is bare soil. Then the squash bugs have nowhere to hide.
I don't till in straw, it doesn't break down enough by spring in my long winter. I rake mine up and put it in the compost pile.