I'm facing the same situation. But I'm new to this too, so don't have a definitive answer. JUst telling you what I'm doing and why. I read on one site that you should avoid putting your run where creeping myrtle is growing. I can move my run *a little* but not completly away from the creeping myrtle. I'm going to assume that roots are poisonous too and do my best to get as much as I can. Then I'm going to cover the area w/ weedblock and keep a close eye out for those little buggers popping up.
I may spray w/ round-up first.
The site I looked at gave 10 or so plants poisonous to chickens, and listed the effect of ingesting it. Creeping myrtle was the most serious, with death being a possible result.
Jon