First of all I would be careful of your boss's opinion as it will be a financial loss to them for you to leave. There is all the costs associated with finding and hiring a replacement to you as well as all the lost opportunity costs of not having an extra body to do work. Thus it is in his interest to keep you around. His counter offer is an example of that. (Especially if he is actually underpaying you - all those potential new hires will be demanding industry standard salaries!)
Secondly. I am aware of Ada but have never used it and have no clue as to where it is used. But with a little google fu I found this site:
Who's Using Ada?
Powered by the Ada Programming Language
While being a couple of years old, you can see that there is a huge number of prominent projects that use Ada in many different fields. It is up to you to decide if you want to work in these fields.
Thirdly. I have been programming various languages for longer than I care to remember. IMHO one of the key things is that no matter what the language, the principles of programming trump the syntax. So becoming skilled in one programming language will always help you in other languages in the future. I also feel that dealing with Ada's reputation for enforcing restrictions will certainly enhance your abilities with languages that are less restrictive, as you will be much more aware of their "sloppiness"
Fourthly. Just because you go down the Ada path doesn't instantly cut you off from the rest of the world. There are many ways you can keep your Java/C# skills current at the same time. For example following things on Stackoverflow, working on OSS projects etc.
Fifthly, working in a language that is considered a niche can be an advantage in the long term. It may not look sexy and up to date, but the sorts of projects that use such languages do so for a good reason yet have only a limited pool of talent to draw from. Hence it makes it potentially more lucrative. For example look at the demand for COBOL programmers right now!
Finally. You obviously want out of your current position as you have been applying for other jobs. I can't tell you if Ada is right for you, or in this new company is reputable or worthwhile, but I can tell you to go with your gut feeling. Just remember that is not a lifetime committment.
BTW check out this question: Learning Ada: Source Code and Newbie “Forum”?