SharePoint Online: You can use your custom domain name, but...

Keep the following things in mind:

1. You cannot use custom domain name in your private (=HTTPS) SharePoint Online sites.
2. What you will can use for your private site depends...you can only hope that https://yourcompanyname.sharepoint.com is available, otherwise you will end up with https://notquitemycompanynamebutthiswillhavetodo.sharepoint.com
3. intra.yourdomain.com CNAME notquitemycompanynamebutthiswillhavetodo.sharepoint.com won't help you either, because you cannot modify AAM settings of your SharePoint Online sites.

What you can do to ease end user's life of they wish to use their custom domain name is to

1. Create public SharePoint Online site with the custom domain name and redirect users to actual private site
   PROs: You can do it with SharePoint Online's Redirect page layout
   CONs: You will use up your one and only public SharePoint Online site slot
2. If you have any other servers with web sites you can host intra.yourdomain.com web site there and have a simple HTTP redirect from that web site to yourcompanyname.sharepoint.com
   PROs: You won't use up public web site slot in SharePoint Online
   CONs: You will need to have web server somewhere where you can implement the HTTP redirection.