The client uses assets.mtgarena.wizards.com to download new card assets, required for updating the client.
That domain has a CNAME record that eventually resolves to a AAAA record, but the server does not properly support IPv6 and drops connections.
When the computer supports IPv6, the connection to that host is done through IPv6, which is not properly implemented, resulting in the client erroring out on the update screen.

To remedy this issue, one of the following should be implemented:
1. The client should implement an explicit fallback to IPv4 when the connection to the IPv6 address fails
2. The server used to serve the update assets should be updated to properly support IPv6

A current workaround is to manually override the DNS resolution of the assets.mtgarena.wizards.com domain to force the connection to this specific host to use IPv4. But this is not a great solution because the IPv4 address could change at any point in the future which would require manual intervention.

Other users have had similar issues (https://feedback.wizards.com/forums/918667-mtg-arena-bugs-product-suggestions/suggestions/49038461-ipv6-not-supported).