Will my flights from USA-Beijing- taiwan be okay? My dates will be January 15-20.

Of course. Your itinerary, if you have mentioned your all stops, is definitely eligible for 24 or 144 hour visa free transit in Beijing. If your transit time in Beijing is less than 24 hours, you will be covered by 24 hour visa free transit and if your transit time is longer than 24 hours for just 1 minute, you will enjoy 144 hour visa free transit. In both cases, you are perfectly fine and you will travel completely without a visa.