It is common to work with proxy when working in a corporate environment.
To work with proxy and React, you must edit the {root project}/android/gradle.properties file:
systemProp.http.proxyHost=url.of.your.proxy.com
systemProp.http.proxyPort=port
systemProp.http.proxyUser=username
systemProp.http.proxyPassword=password
systemProp.https.proxyHost=url.of.your.proxy.com
systemProp.https.proxyPort=port
systemProp.https.proxyUser=username
systemProp.https.proxyPassword=password
You will also need to add the proxy to the nodejs context, in your terminal mac, linux or windows:
npm config set proxy http://user:password@proxy-url:port
npm config set https-proxy http://user:password@proxy-url:port
After that you can run react-native run-android and be happy :D