I have never used PEX, but I understand it will survive freezing water. Is
that correct?

I had a 3/8" soft copper pipe freeze and burst. A plumber put it in a
couple years ago when he installed a dishwasher. There is a dip in it where
it goes under a joist, so when drained my pipes last fall water pooled in it
and froze.
There is no particular reason it shouldn't happen again; I am surprised it
didn't happen the previous two years. However getting to it to work a drain
pipe would be a real pain. I could blow it out with compressed air, but
that wouldn't be so easy either.
Is it reasonable to replace it with PEX; without spending serious money on
tools. I only want one 3' piece. How would I attach it to the 3/8 copper?

Thanks.