No, I can't see why they would terminate you. Having had a good idea, and followed through on that, in the time before you were even employed by them, can't be held against you. 
But they may want to know what you intend to do with that design and IP: If you intend to productize it yourself, or license it to some other company, they may consider that a confict of interest -- for the reasons myself and others have described above. They may want to put some restrictions on that, or a right of first refusal, or they may not care at all. If you are open about it and disclose this early on, that increases the likelyhood that they won't consider it a conflict IMO.
No way I would license it to other companies. Even if I planned on doing it, it'll be very unintelligent for me to tell them that.
The goal of any patent is to make money. I'd make far more money if my employer licensed it and incorporate it into one of their product, since they are a big company with huge resources. Even if they paid me $0.05 on every item, it's a good sum. This is my preferred route.
If they are not interested in licensing it, then I guess I will try to market it myself (actually, have a partner run it since I hate running a business). My main concerns are:
1. Employer decides they are not interested, but also tells me I can't do it on the side (or have someone else run it on the side). This also has the side effect of making them think I'm not dedicated to my job. (not a good start)
2. Employer likes the idea, but wants me to develop it as an employee, which essentially forces me to hand it over. "You develop it here, we'll pay for the utility patent, and you get a "innovator of the year" award."
Maybe I should not bring up the fact that I want to run a side business. Maybe I should just disclose the idea when the agreement asks for it. And then the ball would be in their court to ask me about it. Then I can tell them I hope to eventually license it to them as I believe it's in the interest of the company. I feel this would make them less likely to think I'm not dedicated to my job. It actually make it seem like I have the best interest of the company in heart (kinda true, except I want to make a reasonable royalty as opposed to giving it away).
If they are not interested at the moment, then I can respectfully ask them if I can develop it as a "hobby business" and if it goes well I'd be willing to give them the right to first refusal, while assuring them of my loyalty to the company and my day job.
What do you think?