Does answering "no" to the question "Are you fasting?" while having the intention to fast in one's heart, constitute breaking the fast?

Your fast is valid as long as you do not intend to break it. However, your informing about something contrary to reality is a lie and is impermissible. To avoid showing off, you can use subtle expressions (ma'arid), such as saying "I did not fast" while intending yesterday. This provides an escape from lying.