Can I preserve my Database?

Im working with someone who’s:
•Domain is registered with GoDaddy
•Website WAS hosted with 1&1
•Backend Database is currently on 1&1
•New website is being redirected from GoDaddy to Squarespace

Is there a way I can insert a link into the new website to access the backend database that was associated with the old website eventhough nothing is pointing to the old hosting service ie:Domain Name?

If not is there a way i can copy the whole database?

Below is a picture of the database.

You might consider to copy the whole database.

Log into the old website and dump a copy of the entire database (use mysqldump, etc). Move the dumped file to the new server and load it up.