So sorry to hear about your issues.
If you have trouble connecting to Xbox live.
Please first if there are any service alert, you can check it here: https://beta.support.xbox.com/xbox-live-status
If you cannot connect on Xbox live on backward 360 games using an Xbox One, try the steps below:
A Launch a Backwards Compatible game
B Press the Start and Back (View and Menu) buttons at the same time on the controller.
C Select Manage Storage
D Select Profiles and find your gamertag and delete the gamertag please choose Delete profile only
E Kindly press B to back out of the screen and then press the View and Menu again to bring up the menu
F. Select Sign In or press X on the controller then Select Download Profile to redownload your profile to the console
For Xbox One related connectivity issue.
I would suggest trying to use an manual DNS.
Please follow this steps.
1. Please go to Settings > Network > Advanced Settings > DNS settings >> manual
2. Primary DNS: 208.67.222.222
3. Secondary DNS: 208.67.220.220
Once done, please go back to Settings > power and start up > restart now.
Then try to connect again.
If it still fails, please try to run through the troubleshooting steps on this page for more detailed info: https://support.xbox.com/help/Hardware-Network/…
For Xbox 360 console:
You can go through this guided steps to help you connect: https://beta.support.xbox.com/help/hardware-net…
If you are getting NAT issues, Strict or Moderate.
You need to add and open the follow ports on your router/Modem to allow unrestricted Xbox Live connection on it.
Here are the following ports needed to be unrestricted by Xbox live.
Port 88 (UDP)
Port 3074 (UDP and TCP)
Port 53 (UDP and TCP)
Port 80 (TCP)
Port 500 (UDP)
Port 3544 (UDP)
Port 4500 (UDP)
Please contact your Internet Provider if you are not familiar on the portforwarding procedures.
Hope that helps.