Flash will never die I don't know anyone who wants html5 banners or animations. Also with games. Using direct access to GPU - it will reborn!
No, flash is not dieing. It's mostly adapting to the current market.
Most of what you do in Flash can be done in HTML5/Canvas/JS which explains the new export to Canvas/JS feature in Flash CS6. Banners SHOULD be done in HTML/JS rather than flash for SEO optimization. That has always been Flash's weakness. It wasn't the easiest thing to get good SEO optimization with a full flash website.
Flash is doing 2 things.
1, Adobe is repositioning to be the "Console of the web". They've said this themselves. They're investing in tapping into the GPU since flash player 11. Being the audience of the web, all dutifully with flash player installed, is vastly larger than the console market, they are targeting future high end features to attract game developers to the platform. In fact after August 1st you will need to pay a premium to Adobe if you use its premium features on a "console quality" game and exceed $50,000 in sales.
2, They're discontinuing flash player support for mobile devices (tablets/phones). AIR for devices is taking over here. The flash player was never all that great on my android tablets and phone anyhow. This is probably what your employer heard.
Don't be afraid to use Flash on the web. It's only going to get much better. But use it for its strengths. If you can do it in HTML5/JS then there's no reason to do it in Flash, less you want your site to take a search engine optimization hit and neglect mobile devices in the future.
It's sort of funny Adobe already mentioned discontinued support for Flash Player on mobile devices yet I still get updates for it on my android *Shrug*