Completely delete your account or just the contacts, related to business, that you made while working there?
Yes, they could sue you, but... what would the damages be? In a vacuum, knowing nothing other than what is in your question, sounds like your employer is being unreasonable. This is also assuming your non-compete is valid, which it something that needs to be looked at.
Social media in general is so new, there is not enough significant case law out there to give you a sound answer.
This might come down to you and your old employer finding a middle ground where you delete business related contacts you gained while working there, promise not to contact them for a while, and they covenant not to sue.
You should contact an attorney in your area to talk about all the facts and advise you.
Check out this article written by a law firm about the rules in AZ.
My answer is not intended to be giving legal advice and this topic can be a complex area where the advice of a licensed attorney in your State should be obtained.
It is impossible to tell you if the agreement will hold up without being able to review the agreement and knowing more about the business and your role. Having said all of that, requiring you to delete linkedin connections that you had before you went to work there, or unrelated to your work there are not likley to be allowed as a term in the agreement. It would likely be worth your time to sit down with an attorney for an hour or two and discuss this in greater detail. Good luck.
This information is provided for general informational purposes and is not intended as legal advice. An attorney licensed in your jurisdiction can answer questions specific to your specific fact situation and provide you appropriate advice as necessary based on the specific facts of your matter and the jurisdiction in which you reside. If you are in Arizona and interested in discussing your matter further I can be reached at: (480) 838-9000 Mark D. Fullerton, P.C. 1839 S. Alma School Road, Suite 275 Mesa, Arizona 85210