Intellectual property - Employee Agreement

Go see a lawyer. You're asking for specific legal advice regarding a signed contract. That's what lawyers are for.

how enforceable is this section of the agreement

Depends on several things, including which state law you're subject to, and the specifics of your case. Generally, I think in most of the Western world these provisions are upheld. It is my understanding that California is more lax about this.

how likely is it that a company would enforce this and take ownership?

Totally depends on the people involved, feelings, and the facts of the specific case. In my experience lawsuits against ex-employees are extremely rare, and generally only done when the ex-employee appears to have outright stolen intellectual property or abused his employment relationship. Having said that, there is always the risk that some manager in another division (Human Resource, Legal) decides to start a lawsuit for no good reason.

This question is really hard, because the chances that the company sues can be hard to quantify, and at the same time, if they sue it is highly destructive for your business. If you're just starting out and you get hit by a lawsuit, then 3 things happen:

1. You get lots of legal expenses that you can ill afford.
2. You will be emotionally consumed by the legal proceedings, and will find it hard to do good creative work on your startup.
3. If your customers hear of the lawsuit, they'll run for the hills, i.e. be highly reluctant to do deals with you (so revenue drops).

As I see it, you have 4 main options:

1. Take your chances. Understand the risks, but carry on working on this in your spare time while staying employed with your current employer.
2. Seek an agreement with the company. Tell them what your hopes are, negotiate for a release from the agreement. You could even try to win your employer over as your first costumer if the product could be relevant for them?
3. Take a new job, possibly lower paying, where there are no problematic intellectual property clauses in the contract. Work on your project on the side while employed with this company.
4. Make a clean break. Quit your job before starting your project; start working full-time on your project after you've left the company; be in stealth mode for a couple of months.

I'd like to repeat that you should really see a qualified lawyer who is specialized in this domain of law. Good luck with your decision.

A lot of it depends on your state, but the Evan Brown vs Alcatel case from a while back set a troubling example. From the Register article:

Evan Brown has been working on an idea to convert old computer code so that it can be run on modern machine since 1975.

When Brown mentioned the idea to his employers while he worked for DSC Communications of Plano, Texas (subsequently bought by Alcatel) DSC decided it owned the rights to Brown's insight and demanded that he revealed his idea. Brown refused and he was fired. DSC then launched legal action against him to gain possession of his thoughts.

Sadly, quite a number of employers demand the IP rights to everything you do while under employ, whether it's done on the clock or off the clock. Very few try to enforce it, however.