I don't have Xbox Live and every time I start the game Your Shape Fitness Evolved it says it can't connect to Xbox, so it can't save.
Is there a way to save in Your Shape Fitness Evolved without an Xbox Live account, or is it required?
You do need an Xbox LIVE account. From Ubisoft's FAQ for the game:
As Abby mentioned in the comments, you don't need to pay for the full-blown Xbox LIVE Gold account: the free Xbox LIVE Silver account will do. You can register for a new account either directly on the console by pressing the Xbox button in the center of your controller, and selecting "Join Xbox Live" or via the link Abby mentioned.
It's a dash bug
See solution below
We have identified the issue that is preventing you from playing your Kinect titles unless you are signed in to Xbox LIVE. This issue will be fixed in a future update.
In the meantime, you may work around this issue by setting up an offline profile with Kinect ID enabled. Please review the steps below:
After launching the game, use Kinect gestures to start the game and sign in rather than using a controller. If you still use your controller in the game you may still experience the same issue.
If you have any concerns regarding this notice, you may either reply to this message or contact Xbox Customer Support directly using the information available at http://www.xbox.com/support/contact and reference the Service Request number above.
The Xbox LIVE Escalations Team