Update 10/10/2017

You should try this library instead.

Steps I followed

1. react-native init TodayWidgetTest
2. Added TodayWidget today widget extension target
3. Set TodayViewController as the NSExtensionPrincipalClass in the TodayWidget Info.plist file
4. Removed the storyboard file that came with it
5. Added this code and this code
6. Added App Transport Security Settings in order to Allow Arbitrary Loads in the Info.plist file
7. Under Build Phases, add all React Native binary libraries in the Link Binary with Libraries for the TodayWidget target
8. Under Build Settings, add -ObjC linker flag
9. Set Deployment Target to 7.0

Results

This works on the iOS Simulator, but unfortunately doesn't work on an actual phone.

Related Issue

facebook/react-native#7391