So you may have to seek the advice of a lawyer to understand the particular legal aspects of contract law in your state, for your circumstances, etc.
However, generally speaking you have a contract with your employer which sounds like it makes one of two claims (or both):
Any work you do using company resources (on company time, with company materials and/or company equipment) the company has claim to.
Any work you do using your own resources (on your own time, materials and equipment) they do not have a claim to.
This means that in order for the company to lay claim to your work they have to be able to prove that you violated the contract. You do not have to prove that you did not violate the contract.
So if you use their equipment you can expect there will be some evidence of that use. If you use their materials, it may be noticed that materials went missing. The use of "time" may be difficult to prove either way, but if you use
"company time" and there is evidence of that, then you should be concerned that you will lose partial or full claim to your idea.
The key here is to have separate resources for your idea (computer, equipment, materials, etc.). Also, you should avoid performing any work for your personal idea while on your employer's premises, as this could be considered use of their resources such as "employer paid time" due to your presence there.
Also, if you have a good relationship with your employer then it's possible that you should make your idea known to them (or simply that you have a personal project you are working on outside of work with commercial potential) so that you can discuss the boundaries of your contract. You may also be able to contact someone in HR to express that you have an idea and you want to be sure that you follow your contract appropriately.
What the contract fundamentally does is allow them to have recourse in situations where someone is using company resources for personal gain. In situations where the employer has essentially purchased that employee's time and dedication, and the employee has not provided their time and services as expected, the employer needs recourse. If someone is causing wear on equipment, using parts or supplies from the employer, then the employer is subsidizing the employee's personal endeavors and the company should be compensated.
One of the trickiest points here is that as you pursue this idea, you may find yourself distracted at work. You may also find that you need to take breaks you would not otherwise have taken, or attend to "personal business" during normal business hours. This is typically why employers have these clauses in their contracts, because these distractions may not be significant enough to terminate an employee, but still are a real cost to the company.