Remember, there are lots of different toxic chemicals used to exterminate bugs, so there's no pat answer as to whether extermination on your yard will be safe for your chickens. However, if your exterminator cannot answer your questions about the chemicals he or she is using and whether they are toxic for birds---specifically for birds who'll be grazing the grass---we recommend you find another exterminator who is more knowledgeable and educated. And then never call that FIRST guy, again!

You love your pet chicken, right?
Think about it: if someone is telling you that it will be safe for dogs and cats, that really doesn't mean much in terms of safety for chickens. Dogs and cats don't eat the grass like your chickens do. Plus, any reputable exterminator should know that different animal species react very differently to poisons. For instance, pyrethrins, touted as some of the "safest" chemicals to use for extermination purposes, are pretty easily tolerated by mammals like dogs and cats... but are extraordinarily dangerous for fish and fairly toxic for birds, certainly moreso if your birds are directly eating grass covered with it!
Further, remember that even if your birds don't get sick from ingesting whatever the exterminators are spraying on your lawn, that poison can still be transmitted to you through your birds' eggs, so you need someone who is aware of this and who knows the full chemical impact of their job, inside and out. If you ask about chickens and they tell you about dogs and cats, just RUN, and don't look back. Find a reputable, knowledgeable exterminator that takes the job seriously, and who will answer all your questions so you can make an informed decision.
To reiterate... yuck! We don't think spraying poisons on the same grass your chickens will be eating is a good idea in general, but if you are desperate to get rid of those ants and want to investigate and weigh specific risks, be sure to find someone who can answer your questions in a knowledgeable way and who takes the risk to your family as seriously as you do.