Amazing what they will eat! I would just keep an eye on them. I am pretty sure those things aren't toxic. Mine ate the temperature sensor on the digital thermometer. I think they were sick of getting it up the bum!
I think most fly strips are just wax paper and glue along with an attractant. Shouldn't be harmful...though I would check the label on another container just to be sure.
2 of my bucks somehow managed to get one of those giant fly rolls (1' x 8', stapled to the wall) that I stuck up wayyy high in their barn. They must have really reached up reach it, I put it as high as I could reach.
I found it shredded in their pen the next day.
I have gotten stuck to those things in the past, and it doesn't feel good getting 'unstuck'...and I'm not nearly as hairy as they are.