Unless you hire an industrial hygienist, everyone is grasping at straws. The true source of the irritant may be totally unrelated to the HVAC work or it could be due to some incidental spillage of an irritant into the duct system or accidentally carried into the home.
CO is not an eye and respiratory irritant. Nitrogen dioxide (NO2) is a common respiratory and eye aerosol irritant that is related to combustion byproducts. However, if you are running your AC, it is highly unlikely you would encounter it. If you were burning a ventfree gas heater, I'd say there was a chance. Have you been cooking with a gas oven coincidental to the symptoms?
Again, we can all hypothesize but there is no way of knowing unless a professional, such as an industrial hygienist, is successful at capturing an air sample containing an irritant that does not change into another compound before it can be analyzed. This is very difficult to do with many noxious compounds. If there is a particulate irritant, the chance of achieving a valid sample is much higher.
Sit down with your family and make an honest list of ANY changes in lifestyle: new brand of laundry detergent, carpet cleaning powder, new electronic appliance in the home, etc. You never know what may be the root cause.
If the house remains untenable, contact your insurance carrier and file a claim. That puts the burden on them to:
a) identify the causative agent
b) eradicate the problem
c) take responsibility for ensuring the house is now habitable
If they want to spend a lot of money on environmental testing, let them pay for it. If they think they can identify a culprit, whether the HVAC contractor so someone else, let them worry about possible subrogation against that party. Meanwhile, you get a clean bill of health and a habitable home for the price of your deductible.