If I sign a "Disclosure of Inventions" agreement, does this mean that my employer owns an app I create on my own time?

Probably not, but an experienced attorney would need to see the agreement and the language it contains in full before properly advising you.

Depending on the language, failing to disclose an invention could be considered a breach of the agreement by you. However, your employer's remedy would be to sue you for damages, and they would have the burden of proof in court. If you truly did not use any company time or resources on the invention, then their damages would likely be zero. The company would have to have some reason to believe that you used company time and resources, and the company would have to prove they were damaged by your failure to disclose the invention.