Fist of all Professionalism and Eagle Management are not synonymous terms. That said, I don't see how not responding to an email where someone is asking for a job is not professional. I'm sure they are pretty busy in the HR dept over there. Then again maybe not. I do hear they are having a heck of a time recruiting and holding onto pilots.
Also, if you are (I believe) 40 or older and they give you a hard time, you have an EEOC case on your hands and I would pursue it to the full extent if you really do want to work there. I know an ex-United pilot that got hired at UAL back in the day when Age became a protected class under EEOC law.
However, I probably wouldn't waste your time pursuing an Eagle job. Just my opinion.