You can save Instagram videos on Android phones. You can save the videos in Instagram app. You cannot save it in your phone. Instagram does not provide the facility of saving their any content in phone.
You can save it in app with simple steps:
1. Open the app and log in.
2. Open the video you want to save.
3. Tap the save icon on post in the right corner at bottom.
4. Your video has been saved in app. it becomes black when it save. you can see it in your profile tab.
5. Open your profile tab. tap on the saved icon tab in the right corner above the pictures.
You can un-save the video on tapping it again same when you tap on the save icon.