Last year, Netflix rolled out a server-side UI change that also gets rid of the slide-out menu. It wasn't released as an official update, and it looks like random users are getting it whenever the company wants to install it on their devices. I personally received the UI change over 24 hours ago or so, and as you can see in the screenshots below, its navigation bar is not quite identical to the new one.

[Image credit: Android Police (left and middle) / Mariella Moon (right)]

The original UI change I received a day ago only had four icons in the navigation bar. It has since gotten the "Coming Soon" icon, but has retained My Profile that's been replaced with the More hamburger menu in the new version. If you haven't gotten a server-side update to your UI, your best bet is to join Netflix's beta tester program or sideload the APK courtesy of Android Police.